Groups of order p^8 and exponent p

We prove that for p > 7 there are p^4 + 2p^3 + 20p^2 + 147p + (3p + 29) gcd(p - 1, 3) + 5 gcd(p - 1, 4) + 1246 groups of order p^8 with exponent p. if p is a group of order p^8 and exponent p, and if p has class c > 1 then p is a descendant of p/γ c (p). for each group of exponent p with order less than p^8 we calculate the number of descendants of order p^8 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 p6 and exponent p (p > 3), while we are able to calculate the number of descendants of order p^8, we have not been able to obtain a list of the descendants.