Minimal degrees for faithful permutation representations of groups of order 𝑝⁵ where 𝑝 is an odd prime

Abstract In a 2018 paper, Behravesh and Delfani computed the minimal degree of a faithful permutation representation for groups of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>5</m:mn> </m:msup> </m:math> p^{5} , where 𝑝 is an odd prime. However, their results are incorrect and incomplete: some groups are not considered, some minimal degrees are incorrectly determined, and the reported list of degrees is incomplete. We determine the minimal degree of a faithful permutation representation for each group of order <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>p</m:mi> <m:mn>5</m:mn> </m:msup> </m:math> p^{5} , where 𝑝 is an odd prime. We also record how to obtain such a representation.