Abstract
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 Γ is unsettled. In other words, the classification of the finite groups G such that Cay(G, S) ≅ Γ, for some subset S of G, is still unknown. In this article, we prove that there are at least 2 d 2/64 − (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 n 2/256 − (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 cn 2 , for c = log224.
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