Let $G$ be a finite 2-group of nilpotency class 2 such that $\frac{G}{Z(G)}\simeq C_{2}\times C_{2}$. I want information about its automorphisms group. Please guide me. Thank you

  • $\begingroup$ Is this a homework question, or something else? A first example: D_8. If $G$ is a $2$-group such that $G/Z(G)\cong C_2\times C_2$ the automorphisms are actually quite interesting, as the inner automorphisms act trivially on $Z(G)$ and $G/Z(G)$ even though $Z(G)$ is characteristic (and $\operatorname{Inn}(G)\cong G/Z(G)$)! (I think, but cannot prove, that every outer automorphism is not hidden in this way, which would be quite a nice approach to your problem.) $\endgroup$ – user1729 May 28 '12 at 9:34

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