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I have known cases of abelian and nonabelian groups that have a center of order p but never a power of p. Is there at least a known case where $|Z(G)|=p^2$?

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    How about an abelian group of order $p^2$? If you want non-abelian, you could take the direct product of two non-abelian groups each of which have center of order $p$.2012-09-14
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    So what's the easiest nonabelian example which is not a direct product? How about the group of upper untriangular $3 \times 3$ matrices over the finite field of order $p^2$?2012-09-14
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    You could take a symmetric group $S_n$, which has trivial centre for $n>2$, and is nonabelian, and take a direct product with a group of order $p^2$ (which is necessarily abelian).2012-09-14
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    Keep a lookout for groups with center the Klein $V$ group. They come up a lot.2012-10-16

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