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What is the smallest (in terms of the number of elements) nonabelian group such that any presentation requires at least 3 generators? Most of the nonabelian finite groups I know seem to require only 2 generators.

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Without having thought about it for long, I would guess that the smallest examples have order 16. For example $D_8 \times C_2$ or $Q_8 \times C_2$. They have elementary abelian quotients of order 8, so they definitely need three generators.

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    $\begingroup$ There is only one more rank 3 group of order 16, SmallGroup(16,13), but it doesn't have as nice of a description. $\endgroup$ Commented Jul 26, 2013 at 20:28
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    $\begingroup$ Oh, and these are indeed the smallest examples just by a brute force check. $\endgroup$ Commented Jul 26, 2013 at 20:28
  • $\begingroup$ @JackSchmidt: if you don't mind clarifying, what's $\operatorname{SmallGroup}(m,n)$? I'm having a hard time finding documentation. $\endgroup$ Commented Jul 26, 2013 at 20:56
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    $\begingroup$ @Omnomnomnom It is a command in GAP. The first parameter is the order of the group, the second one is an identifier for which of the groups of that order you want. $\endgroup$ Commented Jul 26, 2013 at 21:02
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    $\begingroup$ According to this website, the other group satisfying these properties is the central product of $D_8$ and $Z_4$ $\endgroup$ Commented Oct 11, 2017 at 18:52

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