There are two nonabelian groups of order p^3, namely, semidirect product of Z/pZ x Z/pZ by Z/pZ and semidirect product of Z/(p^2)Z by Z/pZ. What are the automorphism groups of these groups?
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For the latter group, the answer is Bidwell & Curran's paper "The Automorphism Group of a Split Metacyclic pGroup" 


The former group can be seen as the group of unitriangular $3 \times 3$matrices over the field with $p$ elements: $$G = \\left\{ \begin{pmatrix} 1 & * & * \\\\ 0 & 1 & * \\\\ 0 & 0 & 1 \end{pmatrix} \\right\} \leq SL(3,p)$$ The automorphism groups of such groups have been studied (in a much larger generality); see for instance the paper "The automorphism group of the group of unitriangular matrices over a field" by Ayan Mahalanobis (http://arxiv.org/abs/1012.5534v1). 


The automorphism groups of all pgroups of order p^3 can be found at http://www.math.kth.se/~boij/kandexjobbVT11/Material/pgroups.pdf 

