Isomorphism testing of groups of cube-free order

A group G has cube-free order if no prime to the third power divides |G|. We describe an algorithm that given two cube-free groups G and H of known order, decides whether G≅H, and, if so, constructs an isomorphism G→H. If the groups are input as permutation groups, then our algorithm runs in time polynomial in the input size, improving on the previous super-polynomial bound. An implementation of our algorithm is provided for the computer algebra system GAP.