On finite $p$-groups minimally of class greater than two

Abstract

Let $G$ be a finite nilpotent group of class three whose proper subgroups and proper quotients are nilpotent of class at most two. We show that $G$ is either a 2-generated $p$-group or a 3-generated 3-group. In the first case the groups of maximal order with respect to a given exponent are all isomorphic except in the cases where $p=2$ and $\mathrm {exp}(G)= 2^r$, $r\geq 4$. If $G$ is 3-generated, then we show that there is a unique group of maximal order and exponent 3; but a similar result is not valid for exponent 9.