Finite $p$-groups of class two with a small multiple holomorph

Abstract

We consider the quotient group T (G) of the multiple holomorph by the holomorph of a finite p-group G of class two for an odd prime p. By work of the first-named author, we know that T (G) contains a cyclic subgroup of order p r−1 (p − 1), where p r is the exponent of the quotient of G by its center. In this paper, we shall exhibit examples of G (with r = 1) such that T (G) has order exactly p − 1, which is as small as possible.