Isomorphism in expanding families of indistinguishable groups

Abstract. For every odd prime and every integer , there is a Heisenberg group of order that has pairwise nonisomorphic quotients of order . Yet, these quotients are virtually indistinguishable. They have isomorphic character tables, every conjugacy class of a non-central element has the same size, and every element has order at most . They are also directly and centrally indecomposable and of the same indecomposability type. Nevertheless, there is a polynomial-time algorithm to test for isomorphisms between these groups.