Nilpotency class

Equivalent definitions in tabular format
The notion of nilpotency class (also called nilpotence class) makes sense for any nilpotent group and is a nonnegative integer dependent on the group.

Terminology
When we say that a group has nilpotency class $$c$$, we usually mean that the nilpotency class of the group is at most equal to $$c$$. If we want to say that the class is exactly $$c$$, this is stated explicitly.

Example
Any group of prime power order is nilpotent.

A group of order $$p^n$$, with $$p$$ prime, can have any nilpotency class between 1 and $$n - 1$$ if $$n \ge 2$$. For more information of the number of p-groups of various nilpotency class values for various primes, refer nilpotency class distribution of p-groups.

Relation with derived length
Any nilpotent group is solvable, and there are numerical relations between the nilpotency class and derived length:


 * Derived length is logarithmically bounded by nilpotency class
 * Derived length gives no upper bound on nilpotency class: For a derived length greater than $$1$$, the value of the derived length gives no upper bound on the value of the nilpotency class.

Other related notions

 * Exponent-p class
 * Frattini length
 * Class row of a polynilpotent group