Minimal subgroup with abelianization of maximum order

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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.