Isomorphism in expanding families of indistinguishable groups

For every odd prime p and every integer n >= 12, there is a Heisenberg group of order p(5n/4+Theta(1)) that has pn(2/24+Theta(n)) pairwise nonisomorphic quotients of order p(n). 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 p. 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.