On finite \(p\)-groups minimally of class greater than two (Q1700664)

Summary: 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 \(\exp(G)= 2^r\), \(r\geq 4\). 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.