gap> reps:= List(classes, x-> x[Length(x)]);  # pick 'long' class reps
[ 1, 25, 46, 36, 48, 23, 47, 44 ]
gap> cent:= TransposedMat(xxd * xxd{reps}^-1); # use formula to compute basis of center
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
  [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
  [ 0, -x^22y^22, -x^22y^22, 0, 0, x^20y^20, -x^22y^22, 0, x^16y^22+4x^18y^20+4x^20y^18+x^22y^16, -x^16y^20-2x^18y^18-x^20y^16, -x^16y^20-2x^18y^18-x^20y^16, 0, 0, x^16y^16, 
      -x^16y^20-2x^18y^18-x^20y^16, x^16y^18+x^18y^16, x^14y^16+x^16y^14, 0, 0, 0, -x^12y^14-x^14y^12, x^12y^12, 0, 0, 0, -x^10y^10, -x^10y^10, 0, 0, x^8y^8, -x^10y^10, 0, 
      x^4y^10+4x^6y^8+4x^8y^6+x^10y^4, -x^4y^8-2x^6y^6-x^8y^4, -x^4y^8-2x^6y^6-x^8y^4, 0, 0, x^4y^4, -x^4y^8-2x^6y^6-x^8y^4, x^4y^6+x^6y^4, x^2y^4+x^4y^2, 0, 0, 0, -x^2-y^2, 1, 
      0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x^10y^10, 0, 0, 0, x^8y^8, 0, 0, x^8y^8, 0, -x^4y^6-x^6y^4, -x^4y^6-x^6y^4, x^4y^4, x^4y^4, 0, 0, x^4y^4, 
      x^2y^2+x^4+y^4, -x^2-y^2, -x^2-y^2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, x^20y^20, x^20y^20, -x^18y^20-x^20y^18, 0, x^20y^20, -x^16y^20-3x^18y^18-x^20y^16, 
      0, x^16y^18+x^18y^16, x^16y^16, 0, 0, x^16y^18+x^18y^16, 0, 0, -x^12y^14-x^14y^12, 0, 0, 0, 0, 0, 0, 0, x^8y^10+x^10y^8, x^8y^10+x^10y^8, 0, 0, -x^6y^8-x^8y^6, 0, 0, 
      -2x^4y^8-4x^6y^6-2x^8y^4, x^4y^6+x^6y^4, 2x^4y^6+2x^6y^4, 0, 0, 0, x^4y^6+x^6y^4, 0, -x^2y^2, -x^2-y^2, 0, 0, 1, 0, 0, 1 ], 
  [ 0, 0, 0, 0, 0, 0, 0, 0, -x^4y^6-x^6y^4, x^4y^4, x^4y^4, 0, 0, 0, x^4y^4, 0, -2x^2-2y^2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
      0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x^4y^6-x^6y^4, x^4y^4, x^4y^4, 0, 0, 0, x^4y^4, 0, 
      -2x^2-2y^2, 1, 1, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, x^16y^20+2x^18y^18+x^20y^16, -x^16y^18-x^18y^16, -x^16y^18-x^18y^16, 0, x^16y^16, 0, -x^16y^18-x^18y^16, 
      x^16y^16, x^12y^16+2x^14y^14+x^16y^12, -x^12y^14-x^14y^12, -x^12y^14-x^14y^12, x^12y^12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, x^4y^8+2x^6y^6+x^8y^4, -x^4y^6-x^6y^4, 
      -x^4y^6-x^6y^4, 0, x^4y^4, 0, -x^4y^6-x^6y^4, x^4y^4, 2x^2y^2+x^4+y^4, -x^2-y^2, -x^2-y^2, 1, 0, 0, 0, 0 ] ]
gap> set:= Set(Flat(cent));  # collect all distinct coefficients
[ 0, -2x^4y^8-4x^6y^6-2x^8y^4, -2x^2-2y^2, -x^2y^2, -x^10y^10, -x^22y^22, -x^16y^20-3x^18y^18-x^20y^16, -x^4y^8-2x^6y^6-x^8y^4, -x^16y^20-2x^18y^18-x^20y^16, -x^2-y^2, 
  -x^4y^6-x^6y^4, -x^6y^8-x^8y^6, -x^12y^14-x^14y^12, -x^16y^18-x^18y^16, -x^18y^20-x^20y^18, 1, x^4y^4, x^8y^8, x^12y^12, x^16y^16, x^20y^20, x^2y^4+x^4y^2, x^4y^6+x^6y^4, 
  x^8y^10+x^10y^8, x^14y^16+x^16y^14, x^16y^18+x^18y^16, x^2y^2+x^4+y^4, x^4y^8+2x^6y^6+x^8y^4, x^12y^16+2x^14y^14+x^16y^12, x^16y^20+2x^18y^18+x^20y^16, 
  x^4y^10+4x^6y^8+4x^8y^6+x^10y^4, x^16y^22+4x^18y^20+4x^20y^18+x^22y^16, 2x^2y^2+x^4+y^4, 2x^4y^6+2x^6y^4 ]
gap> denom:= Set(List(set, x-> x.den)); # collect their denominators
[ 1 ]
gap> classes[4];
[ 17, 21, 24, 28, 29, 32, 33, 36 ]
gap> reps[4]:= 33;  # pick a different rep for class 4
33
gap> cent:= TransposedMat(xxd * xxd{reps}^-1); # use formula to compute basis of center
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
  [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
  [ 0, -x^22y^22, -x^22y^22, 0, 0, x^20y^20, -x^22y^22, 0, x^16y^22+4x^18y^20+4x^20y^18+x^22y^16, -x^16y^20-2x^18y^18-x^20y^16, -x^16y^20-2x^18y^18-x^20y^16, 0, 0, x^16y^16, 
      -x^16y^20-2x^18y^18-x^20y^16, x^16y^18+x^18y^16, (2x^14y^20+6x^16y^18+6x^18y^16+2x^20y^14)/(x^2y^2+x^4+y^4), 0, 0, 0, 
      (-2x^12y^18-6x^14y^16-6x^16y^14-2x^18y^12)/(x^2y^2+x^4+y^4), x^12y^12, 0, (-x^12y^18-4x^14y^16-4x^16y^14-x^18y^12)/(x^2y^2+x^4+y^4), 0, 
      (x^8y^16+4x^10y^14+7x^12y^12+4x^14y^10+x^16y^8)/(x^2y^2+x^4+y^4), (x^8y^16+4x^10y^14+7x^12y^12+4x^14y^10+x^16y^8)/(x^2y^2+x^4+y^4), 
      (-x^8y^14-4x^10y^12-4x^12y^10-x^14y^8)/(x^2y^2+x^4+y^4), (-x^8y^14-4x^10y^12-4x^12y^10-x^14y^8)/(x^2y^2+x^4+y^4), x^8y^8, -x^10y^10, 
      (-x^8y^14-4x^10y^12-4x^12y^10-x^14y^8)/(x^2y^2+x^4+y^4), 0, (2x^6y^10+4x^8y^8+2x^10y^6)/(x^2y^2+x^4+y^4), (2x^6y^10+4x^8y^8+2x^10y^6)/(x^2y^2+x^4+y^4), 
      (-x^4y^10-4x^6y^8-4x^8y^6-x^10y^4)/(x^2y^2+x^4+y^4), 0, x^4y^4, -x^4y^8-2x^6y^6-x^8y^4, x^4y^6+x^6y^4, x^2y^4+x^4y^2, 0, 0, 0, -x^2-y^2, 1, 0, 0 ], 
  [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -x^10y^10/(x^2y^2+x^4+y^4), 0, 0, 0, x^8y^8/(x^2y^2+x^4+y^4), 0, 0, x^8y^8/(x^2y^2+x^4+y^4), 0, 
      (-x^4y^6-x^6y^4)/(x^2y^2+x^4+y^4), (-x^4y^6-x^6y^4)/(x^2y^2+x^4+y^4), x^4y^4/(x^2y^2+x^4+y^4), x^4y^4/(x^2y^2+x^4+y^4), 0, 0, x^4y^4/(x^2y^2+x^4+y^4), 1, 
      (-x^2-y^2)/(x^2y^2+x^4+y^4), (-x^2-y^2)/(x^2y^2+x^4+y^4), 1/(x^2y^2+x^4+y^4), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
  [ 0, 0, 0, x^20y^20, x^20y^20, -x^18y^20-x^20y^18, 0, x^20y^20, -x^16y^20-3x^18y^18-x^20y^16, 0, x^16y^18+x^18y^16, x^16y^16, 0, 0, x^16y^18+x^18y^16, 0, 
      (-2x^14y^18-4x^16y^16-2x^18y^14)/(x^2y^2+x^4+y^4), -x^12y^14-x^14y^12, 0, 0, (2x^12y^16+4x^14y^14+2x^16y^12)/(x^2y^2+x^4+y^4), 0, 0, 
      (2x^12y^16+4x^14y^14+2x^16y^12)/(x^2y^2+x^4+y^4), 0, (-x^8y^14-4x^10y^12-4x^12y^10-x^14y^8)/(x^2y^2+x^4+y^4), (-x^8y^14-4x^10y^12-4x^12y^10-x^14y^8)/(x^2y^2+x^4+y^4), 
      (2x^8y^12+4x^10y^10+2x^12y^8)/(x^2y^2+x^4+y^4), (2x^8y^12+4x^10y^10+2x^12y^8)/(x^2y^2+x^4+y^4), -x^6y^8-x^8y^6, 0, (2x^8y^12+4x^10y^10+2x^12y^8)/(x^2y^2+x^4+y^4), 0, 
      (-x^4y^10-4x^6y^8-4x^8y^6-x^10y^4)/(x^2y^2+x^4+y^4), (-2x^6y^8-2x^8y^6)/(x^2y^2+x^4+y^4), (2x^4y^8+4x^6y^6+2x^8y^4)/(x^2y^2+x^4+y^4), 0, 0, x^4y^6+x^6y^4, 0, -x^2y^2, 
      -x^2-y^2, 0, 0, 1, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 0, 0, -x^4y^6-x^6y^4, x^4y^4, x^4y^4, 0, 0, 0, x^4y^4, 0, -2x^2-2y^2, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 
      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (-x^14y^16-x^16y^14)/(x^2y^2+x^4+y^4), 0, 0, 0, 
      (x^12y^14+x^14y^12)/(x^2y^2+x^4+y^4), 0, 0, (x^12y^14+x^14y^12)/(x^2y^2+x^4+y^4), 0, (-x^8y^12-2x^10y^10-x^12y^8)/(x^2y^2+x^4+y^4), 
      (-x^8y^12-2x^10y^10-x^12y^8)/(x^2y^2+x^4+y^4), (x^8y^10+x^10y^8)/(x^2y^2+x^4+y^4), (x^8y^10+x^10y^8)/(x^2y^2+x^4+y^4), 0, 0, (x^8y^10+x^10y^8)/(x^2y^2+x^4+y^4), 0, 
      -x^6y^6/(x^2y^2+x^4+y^4), -x^6y^6/(x^2y^2+x^4+y^4), (x^4y^6+x^6y^4)/(x^2y^2+x^4+y^4), 0, 0, x^4y^4, 0, -2x^2-2y^2, 1, 1, 0, 0, 0, 1, 0 ], 
  [ 0, 0, 0, 0, 0, 0, 0, 0, x^16y^20+2x^18y^18+x^20y^16, -x^16y^18-x^18y^16, -x^16y^18-x^18y^16, 0, x^16y^16, 0, -x^16y^18-x^18y^16, x^16y^16, 
      (x^12y^20+4x^14y^18+6x^16y^16+4x^18y^14+x^20y^12)/(x^2y^2+x^4+y^4), -x^12y^14-x^14y^12, -x^12y^14-x^14y^12, x^12y^12, (-x^12y^16-2x^14y^14-x^16y^12)/(x^2y^2+x^4+y^4), 0, 
      0, (-x^12y^16-2x^14y^14-x^16y^12)/(x^2y^2+x^4+y^4), 0, (x^8y^14+3x^10y^12+3x^12y^10+x^14y^8)/(x^2y^2+x^4+y^4), (x^8y^14+3x^10y^12+3x^12y^10+x^14y^8)/(x^2y^2+x^4+y^4), 
      (-x^8y^12-2x^10y^10-x^12y^8)/(x^2y^2+x^4+y^4), (-x^8y^12-2x^10y^10-x^12y^8)/(x^2y^2+x^4+y^4), 0, 0, (-x^8y^12-2x^10y^10-x^12y^8)/(x^2y^2+x^4+y^4), 0, 
      (x^6y^8+x^8y^6)/(x^2y^2+x^4+y^4), (x^6y^8+x^8y^6)/(x^2y^2+x^4+y^4), (-x^4y^8-2x^6y^6-x^8y^4)/(x^2y^2+x^4+y^4), x^4y^4, 0, -x^4y^6-x^6y^4, x^4y^4, 2x^2y^2+x^4+y^4, 
      -x^2-y^2, -x^2-y^2, 1, 0, 0, 0, 0 ] ]
gap> set:= Set(Flat(cent));  # collect all distinct coefficients
[ 0, -2x^2-2y^2, -x^2y^2, -x^10y^10, -x^22y^22, -x^16y^20-3x^18y^18-x^20y^16, -x^4y^8-2x^6y^6-x^8y^4, -x^16y^20-2x^18y^18-x^20y^16, -x^2-y^2, -x^4y^6-x^6y^4, -x^6y^8-x^8y^6, 
  -x^12y^14-x^14y^12, -x^16y^18-x^18y^16, -x^18y^20-x^20y^18, 1, x^4y^4, x^8y^8, x^12y^12, x^16y^16, x^20y^20, x^2y^4+x^4y^2, x^4y^6+x^6y^4, x^16y^18+x^18y^16, 
  x^16y^20+2x^18y^18+x^20y^16, x^16y^22+4x^18y^20+4x^20y^18+x^22y^16, 2x^2y^2+x^4+y^4, (-2x^12y^18-6x^14y^16-6x^16y^14-2x^18y^12)/(x^2y^2+x^4+y^4), 
  (-2x^14y^18-4x^16y^16-2x^18y^14)/(x^2y^2+x^4+y^4), (-2x^6y^8-2x^8y^6)/(x^2y^2+x^4+y^4), -x^6y^6/(x^2y^2+x^4+y^4), -x^10y^10/(x^2y^2+x^4+y^4), 
  (-x^4y^10-4x^6y^8-4x^8y^6-x^10y^4)/(x^2y^2+x^4+y^4), (-x^8y^14-4x^10y^12-4x^12y^10-x^14y^8)/(x^2y^2+x^4+y^4), (-x^12y^18-4x^14y^16-4x^16y^14-x^18y^12)/(x^2y^2+x^4+y^4), 
  (-x^4y^8-2x^6y^6-x^8y^4)/(x^2y^2+x^4+y^4), (-x^8y^12-2x^10y^10-x^12y^8)/(x^2y^2+x^4+y^4), (-x^12y^16-2x^14y^14-x^16y^12)/(x^2y^2+x^4+y^4), (-x^2-y^2)/(x^2y^2+x^4+y^4), 
  (-x^4y^6-x^6y^4)/(x^2y^2+x^4+y^4), (-x^14y^16-x^16y^14)/(x^2y^2+x^4+y^4), 1/(x^2y^2+x^4+y^4), x^4y^4/(x^2y^2+x^4+y^4), x^8y^8/(x^2y^2+x^4+y^4), (x^4y^6+x^6y^4)/(x^2y^2+x^4+y^4)
    , (x^6y^8+x^8y^6)/(x^2y^2+x^4+y^4), (x^8y^10+x^10y^8)/(x^2y^2+x^4+y^4), (x^12y^14+x^14y^12)/(x^2y^2+x^4+y^4), (x^8y^14+3x^10y^12+3x^12y^10+x^14y^8)/(x^2y^2+x^4+y^4), 
  (x^12y^20+4x^14y^18+6x^16y^16+4x^18y^14+x^20y^12)/(x^2y^2+x^4+y^4), (x^8y^16+4x^10y^14+7x^12y^12+4x^14y^10+x^16y^8)/(x^2y^2+x^4+y^4), (2x^4y^8+4x^6y^6+2x^8y^4)/(x^2y^2+x^4+y^4)
    , (2x^6y^10+4x^8y^8+2x^10y^6)/(x^2y^2+x^4+y^4), (2x^8y^12+4x^10y^10+2x^12y^8)/(x^2y^2+x^4+y^4), (2x^12y^16+4x^14y^14+2x^16y^12)/(x^2y^2+x^4+y^4), 
  (2x^14y^20+6x^16y^18+6x^18y^16+2x^20y^14)/(x^2y^2+x^4+y^4) ]
gap> denom:= Set(List(set, x-> x.den)); # collect their denominators
[ 1, x^2y^2+x^4+y^4 ]
gap> LogTo();