Definition 6

A group $G$ is called nilpotent if there exists an integer $n$ such that $C^{n+1}(G) = \{e\}$. The least integer $n$ such that $C^{n+1}(G) = \{e\}$ is called the nilpotency class of a nilpotent group $G$.

If $n \in \mathbf{N}$, a group of nilpotency class $n$ is called a nilpotent group of class $n$. It is sometimes said that the nilpotency class of a group $G$ is finite if $G$ is nilpotent.