A group that can be written as $G/Z(G)$ for some group $G$ is called capable. Can someone list the capable groups of order 32?

  • $\begingroup$ What did you tried to do? Do you know which groups of order $32$ exist at all? $\endgroup$ – Ofir Schnabel Jan 8 '15 at 7:58
  • $\begingroup$ I want to know if the following groups of order 32 are capable? smallgroup(32,6), smallgroup(32,7), smallgroup(32,8), smallgroup(32,43), smallgroup(32,44). Also we know that $D_{32}$ is capable. $\endgroup$ – M. R. Jan 8 '15 at 9:00
  • $\begingroup$ The quaternion group $Q_{32}$ is not capable. For criteria see also here. $\endgroup$ – Dietrich Burde Jan 8 '15 at 10:04
  • $\begingroup$ @ Dietrich Burde: What about the other groups? $\endgroup$ – M. R. Sep 2 '15 at 6:28

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