Minimal subgroup with abelianization of maximum order

Definition

Let $P$ be a group of prime power order. A subgroup $B$ of $P$ is termed a minimal subgroup with abelianization of maximum order if $B$ is a subgroup with abelianization of maximum order and no proper subgroup of $B$ is a subgroup with abelianization of maximum order.

Any such subgroup is a group of nilpotency class two, because group of prime power order having a larger abelianization than any proper subgroup has class two.