Example

(1) A group is nilpotent of class 0 (resp. $\leq 1$) if and only if it consists of the identity element (resp. is commutative).

(2) For every commutative ring $A$ and every integer $n \geq 1$, the upper strict triangular group $T_1(n, A)$ is nilpotent of class $\leq n - 1$ (and exactly of class $n - 1$ if $A \neq \{0\}$).

(3) Let $G$ be a nilpotent group of class $n$. Every subgroup (resp. every quotient group) of $G$ is nilpotent of class $\leq n$. For, if $H$ is a subgroup of $G$, then $C^n(H) \subset C^n(G)$. If $G'$ is a quotient group of $G$ and $\pi : G \to G'$ is the canonical homomorphism, then $C^n(G') = \pi(C^n(G))$.

(4) A finite product of nilpotent groups is nilpotent.