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The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter

Definition

A maximal class group is a group of prime power order that has nilpotency class c, and order p^{c + 1}, where p is a prime number and c > 1. Equivalently, it has the following properties:

1. The abelianization of the group, i.e., the quotient of the group by its commutator subgroup has order p².
2. The upper central series and lower central series coincide, and all the successive quotients (except the top-most quotient) are of order p.

Relation with other properties

Weaker properties

• UL-equivalent group