Minimal degrees for faithful permutation representations of groups of order $p^6$ where $p$ is an odd prime

TL;DR

This paper determines the smallest degree of faithful permutation representations for all groups of order p^6 with p an odd prime, providing explicit methods to construct these representations.

Contribution

It explicitly computes minimal degrees for all such groups and details how to obtain the corresponding faithful permutation representations.

Findings

Minimal degrees for groups of order p^6 are fully classified.

Explicit construction methods for faithful permutation representations are provided.

Results apply to all odd primes p, offering a comprehensive understanding.

Abstract

We determine the minimal degree of a faithful permutation representation for each group of order $p^6$ where $p$ is an odd prime. We also record how to obtain such a representation.