"

As near as I can tell, wikipedia has no article on ""cycle structure"". The cycle structure of an element g of a permutation group is the multi-set of sizes of orbits of the cyclic group generated by g. This is related to the cycle decomposition of g when g only has finite orbits, which takes orbit representatives a1, a2, ... and specifies g by (a1, a1^g, a1^(g^2), ... )(a2, a2^g, ...) where each tuple has no duplicate entries. This is also called ""cycle notation"", and the ""cycle structure"" is the list of cycle sizes.
For GAP, you can handle much larger groups using its character table library.  The following gives all the sizes, orders, and power relations for the O'Nan group. To check that 8a and 8b are not power conjugate, use the last commands.
 gap> LoadPackage(""ctbllib"");;
 # Get the O'Nan group as a character table
 gap> ct = CharacterTable(""ON"");;
 gap> OrdersClassRepresentatives(ct);
 gap> SizesConjugacyClasses(ct);
 # See which classes are 13th powers of each other
 gap> List( Filtered(
 > Cycles(PermList(PowerMap(ct,13)), [1..NrConjugacyClasses(ct)]),
 > x->Length(x)>1), x->ClassNames(ct){x});
 [ [ ""15a"", ""15b"" ], [ ""16a"", ""16b"" ], [ ""16c"", ""16d"" ], [ ""19a"", ""19c"", ""19b"" ],
   [ ""20a"", ""20b"" ], [ ""28a"", ""28b"" ], [ ""31a"", ""31b"" ] ]
 gap> PrintArray(TransposedMat([ClassNames(ct), 
 > OrdersClassRepresentatives(ct),
 > SizesConjugacyClasses(ct),
 > List(SizesConjugacyClasses(ct),StringPP)]));
 [[  1a,  1,           1,                       1 ],                                                             
  [  2a,  2,     2857239,        3^2*7^2*11*19*31 ],                                                             
  [  3a,  3,   142227008,        2^6*7^3*11*19*31 ],                                                             
  [  4a,  4,     5714478,      2*3^2*7^2*11*19*31 ],                                                             
  [  4b,  4,  1800060570,    2*3^4*5*7^3*11*19*31 ],                                                             
  [  5a,  5,  2560086144,    2^7*3^2*7^3*11*19*31 ],                                                             
  [  6a,  6,  6400215360,  2^6*3^2*5*7^3*11*19*31 ],                                                             
  [  7a,  7,   335871360,      2^7*3^4*5*11*19*31 ],                                                             
  [  7b,  7,  9404398080,    2^9*3^4*5*7*11*19*31 ],                                                             
  [  8a,  8, 14400484560,  2^4*3^4*5*7^3*11*19*31 ],                                                             
  [  8b,  8, 14400484560,  2^4*3^4*5*7^3*11*19*31 ],                                                             
  [ 10a, 10, 23040775296,    2^7*3^4*7^3*11*19*31 ],                                                             
  [ 11a, 11, 41892318720,     2^9*3^4*5*7^3*19*31 ],                                                             
  [ 12a, 12, 12800430720,  2^7*3^2*5*7^3*11*19*31 ],                                                             
  [ 14a, 14, 16457696640,  2^7*3^4*5*7^2*11*19*31 ],                                                             
  [ 15a, 15, 10240344576,    2^9*3^2*7^3*11*19*31 ],                                                             
  [ 15b, 15, 10240344576,    2^9*3^2*7^3*11*19*31 ],                                                             
  [ 16a, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],                                                             
  [ 16b, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],                                                             
  [ 16c, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],                                                             
  [ 16d, 16, 28800969120,  2^5*3^4*5*7^3*11*19*31 ],                                                             
  [ 19a, 19, 24253447680,     2^9*3^4*5*7^3*11*31 ],                                                             
  [ 19b, 19, 24253447680,     2^9*3^4*5*7^3*11*31 ],                                                             
  [ 19c, 19, 24253447680,     2^9*3^4*5*7^3*11*31 ],                                                             
  [ 20a, 20, 23040775296,    2^7*3^4*7^3*11*19*31 ],                                                             
  [ 20b, 20, 23040775296,    2^7*3^4*7^3*11*19*31 ],                                                             
  [ 28a, 28, 16457696640,  2^7*3^4*5*7^2*11*19*31 ],                                                             
  [ 28b, 28, 16457696640,  2^7*3^4*5*7^2*11*19*31 ],                                                             
  [ 31a, 31, 14865016320,     2^9*3^4*