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.