The class three groups of order $p^9$ with exponent $p$

TL;DR

This paper classifies certain p-groups of order p^9 with exponent p, showing that the number of their descendants follows a PORC pattern, but some cases exhibit non-PORC behavior.

Contribution

It provides a complete classification of class two groups of order dividing p^8 with exponent p and analyzes the PORC nature of their descendants.

Findings

Number of descendants is PORC for most cases.

Some groups have non-PORC number of descendants.

Complete classification of specific p-groups.

Abstract

We give a complete list of the class two groups with exponent $p$ and order dividing $p^8$. For each group in the list we compute the number of immediate descendants of order $p^9$ with exponent $p$. In each case the number of descendants is PORC, and so the total number of class three groups of order $p^9$ with exponent $p$ is PORC. Nevertheless, there are groups of order $p^8$ with exponent $p$ which have a non-PORC number of class three descendants of order $p^9$ with exponent $p$.