ENUMERATING GROUPS ACTING REGULARLY ON A d-DIMENSIONAL CUBE

In the survey article [2] it was noted, among many other open problems, that the classification of the groups acting regularly on a d-dimensional cube Gamma is unsettled. In other words, the classification of the finite groups G such that Cay(G, S) congruent to Gamma, for some subset S of G, is still unknown. In this article, we prove that there are at least 2(d2/64-(d/2)log2(d/2)) nonisomorphic 2-groups of Frattini class 2 acting regularly on a d-dimensional cube. Other relevant results are presented. As a corollary of our result, we remark that the symmetric group Sym(n) on n symbols contains at least 2(n2/256-(n/4)log2(n/4)) subgroups up to isomorphism. In particular, we recall that in [4] it was proved that the total number of subgroups of Sym(n) is at most 2(cn2), for c = log(2) 24.