TITLE: Is there a way of canonically labelling permutation groups?
QUESTION [21 upvotes]: When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with identity checks.
Currently I am working with large numbers of transitive permutation groups; the degree is small (40ish) but the size of group can be large, and my programs are gumming up due to having to perform large numbers of - not exactly isomorphism - but conjugacy checks. (I need to check each pair of groups to see if they are conjugate in the symmetric group.) 
This process would be considerably easier if there was a known way of finding a canonical conjugate of the incoming permutation groups...

REPLY [2 votes]: In GAP, for transitive permutation groups of degree at most 30 one can use  TransitiveIdentification(G) from the package TransGrp by Alexander Hulpke. This gives the index of the (unique!) permutation group in the library that is conjugate to the given permutation group and it is apparently quite fast. See section 13 of 
Hulpke, Alexander, Constructing transitive permutation groups., J. Symb. Comput. 39, No. 1, 1-30 (2005). ZBL1131.20003 MR2168238.