  For every odd prime $p$ and every integer $n\geq 12$ there is a Heisenberg
group of order $p^{5n/4+O(1)}$ that has $p^{n^2/24+O(n)}$ 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. The recognized portions of their automorphism groups
are isomorphic, represented isomorphically on their abelianizations, and of
small index in their full automorphism groups. Nevertheless, there is a
polynomial-time algorithm to test for isomorphisms between these groups.
