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  "body": "\n## Definition\n\nLet $G$ be a [[nilpotent group|0Q7MMQHC1P]].\nLet $\\gamma_1(G) = G$ and $\\gamma_{i+1}(G) = [\\gamma_i(G), G]$ be its [[lower central series|0K98R7A4VC]].\nThe *nilpotency class* of $G$ is the least integer $c \\ge 0$ such that $\\gamma_{c+1}(G) = \\{e\\}$.\n\n## Remarks\n\nEquivalently, the class is the least $c$ such that the [[upper central series|0XW54DH28V]] satisfies $Z_c(G) = G$.\nIt is also the least length of a central series of $G$.\nThe trivial group has class $0$, and a nontrivial [[abelian group|082GHNPJ53]] has class $1$.\nExample: the [[quaternion group|0NRJGW864D]] $Q_8$ has class $2$.\nIndeed $\\gamma_2(Q_8) = [Q_8, Q_8] = \\{\\pm 1\\} \\neq \\{1\\}$, and $\\gamma_3(Q_8) = [\\{\\pm 1\\}, Q_8] = \\{1\\}$ because $-1$ is central.\nNon-example: the symmetric group $S_3$ has no class, because it is not nilpotent.\nIts lower central series is $\\gamma_1 = S_3$ and then $\\gamma_i = A_3$ for all $i \\ge 2$, so it never reaches $\\{e\\}$.\n\n## Explanations\n\n### L1 human\n\nThe class counts how many times you must take commutators with the whole group before everything becomes trivial.\n",
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