Groups of order p^۸ and exponent p

‎We prove that for p>۷ there are ‎\[‎ ‎p^{۴}+۲p^{۳}+۲۰p^{۲}+۱۴۷p+(۳p+۲۹)\gcd (p-۱,۳)+۵\gcd (p-۱,۴)+۱۲۴۶‎ ‎\] ‎groups of order p^{۸} with exponent p‎. ‎If P is a group of order p^{۸}‎ ‎and exponent p‎, ‎and if P has class c>۱ then P is a descendant of ‎P/\gamma _{c}(P)‎. ‎For each group of exponent p with order less than ‎p^{۸} we calculate the number of descendants of order p^{۸} with‎ ‎exponent p‎. ‎In all but one case we are able to obtain a complete and‎ ‎irredundant list of the descendants‎. ‎But in the case of the three generator‎ ‎class two group of order p^{۶} and exponent p (p>۳)‎, ‎while we are able‎ ‎to calculate the number of descendants of order p^{۸}‎, ‎we have not been‎ ‎able to obtain a list of the descendants‎.

Keywords:

Groups of exponent p , Groups of order p^۸ , PORC conjecture