Minimal degrees for faithful permutation representations of groups of order \(p^6\) where \(p\) is an odd prime (Q6618156)

Let \(G\) be a finite group, the minimal faithful permutation degree \(\mu(G)\) of \(G\) is the least positive integer \(n\) such that \(G\) is isomorphic to a subgroup of the symmetric group \(S_{n}\), that is, \(\mu(G)\) is the minimal degree of a faithful representation of \(G\) by permutation matrices. A quasi-permutation matrix is a square matrix over the complex field \(\mathbb{C}\) with non-negative integral trace. If \(c(G)\) is the minimal degree of a faithful quasi-permutation representation of \(G\), then \(c(G) \leq \mu(G)\).\N\NIn the paper under review, the authors compute \(\mu(G)\) for the non-abelian groups of order \(p^{6}\) where \(p\) is an odd prime. They provide a database of parameterized presentations for these groups (available publicly in \textsf{GAP} and \textsc{Magma}). The authors also obtain important information on \(c(G)\), always in the case when \(G\) is non-abelian and \(|G|=p^{6}\) (\(p>2\)).\N\NThe reviewer points out that a list of faithful representations for the \(267\) groups of order dividing \(2^{6}\) can be found in [\textit{M. Hall jun.} and \textit{J. K. Senior}, The groups of order \(2^{n}\) (\(n\leq 6\)). New York: The Macmillan Company; Toronto, Ontario: Collier-Macmillan Canada (1964; Zbl 0192.11701)] (with some minor inaccuracies corrected in later papers).