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 a a and specifies g by a a g a g a a 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 a and b 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 th powers of each other gap list filtered cycles permlist powermap ct nrconjugacyclasses ct x length x x classnames ct x a b a b c d a c b a b a b a b gap printarray transposedmat classnames ct ordersclassrepresentatives ct sizesconjugacyclasses ct list sizesconjugacyclasses ct stringpp 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