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* For a small non-abelian example, consider the [[quaternion group]] ''Q''<sub>8</sub>, which is a smallest non-abelian ''p''-group. It has center {1, −1} of order 2, and its upper central series is {1}, {1, −1}, ''Q''<sub>8</sub>; so it is nilpotent of class 2.
* The [[direct product]] of two nilpotent groups is nilpotent.<ref>{{cite book|author=Zassenhaus |title=The theory of groups|year=1999|url={{Google books|plainurl=y|id=eCBK6tj7_vAC|page=143|text=The direct product of a finite number of nilpotent groups is nilpotent}}|page=143}}</ref>
* All finite [[p-group|''p''-group]]s are in fact nilpotent ([[p-group#Non-trivial center|proof]]). The maximal class of a group of order ''p''<sup>''n''</sup> is ''n'' (
* Furthermore, every finite nilpotent group is the direct product of ''p''-groups.<ref>{{cite book|author=Zassenhaus|booktitle=The theory of groups|year=1999|url={{Google books|plainurl=y|id=eCBK6tj7_vAC|page=143|text=Every finite nilpotent group is the direct product of its Sylow groups}}|page=143|title=Theorem 11}}</ref>
* The [[Heisenberg group]] ''H'' is an example of non-abelian,<ref>{{cite book|author=Haeseler |title=Automatic Sequences (De Gruyter Expositions in Mathematics, 36)|year=2002|url={{Google books|plainurl=y|id=wmh7tc6uGosC|page=15|text=The Heisenberg group is a non-abelian}}|page=15}}</ref> infinite nilpotent group.<ref>{{cite book|author=Palmer |title= Banach algebras and the general theory of *-algebras|year=2001|url={{Google books|plainurl=y|id=zn-iZNNTb-AC|page=1283|text=Heisenberg group this group has nilpotent length 2 but is not abelian}}|page=1283}}</ref> It has nilpotency class 2 with central series 1, ''Z''(''H''), ''H''.
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