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 multiset 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 a a  and specifies g by a ag ag  a ag  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 ONan group To check that a and b are not power conjugate use the last commands  gap LoadPackagectbllib   Get the ONan group as a character table  gap ct  CharacterTableON  gap OrdersClassRepresentativesct  gap SizesConjugacyClassesct   See which classes are th powers of each other  gap List Filtered   CyclesPermListPowerMapct NrConjugacyClassesct   xLengthx xClassNamesctx    a b   a b   c d   a c b      a b   a b   a b    gap PrintArrayTransposedMatClassNamesct    OrdersClassRepresentativesct   SizesConjugacyClassesct   ListSizesConjugacyClassesctStringPP    a                                                                                                       a                                                                                  a                                                                                a                                                                                b                                                                           a                                                                           a                                                                         a                                                                              b                                                                           a                                                                        b                                                                       a                                                                        a                                                                         a                                                                      a                                                                      a                                                                        b                                                                        a                                                                      b                                                                      c                                                                      d                                                                      a                                                                         b                                                                         c                                                                         a                                                                        b                                                                        a                                                                      b                                                                      a