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