Nilpotency class

Remarks

Equivalently, the class is the least $c$ such that the upper central series satisfies $Z_c(G) = G$. It is also the least length of a central series of $G$. The trivial group has class $0$, and a nontrivial abelian group has class $1$. Example: the quaternion group $Q_8$ has class $2$. Indeed $\gamma_2(Q_8) = [Q_8, Q_8] = \{\pm 1\} \neq \{1\}$, and $\gamma_3(Q_8) = [\{\pm 1\}, Q_8] = \{1\}$ because $-1$ is central. Non-example: the symmetric group $S_3$ has no class, because it is not nilpotent. Its lower central series is $\gamma_1 = S_3$ and then $\gamma_i = A_3$ for all $i \ge 2$, so it never reaches $\{e\}$.