GROUPS OF ORDER p8 AND EXPONENT p.

We prove that for p > 7 there are p4 + 2p3 + 20p2 + 147p + (3p + 29) gcd(p - 1; 3) + 5 gcd(p - 1; 4) + 1246 groups of order p8 with exponent p. If P is a group of order p8 and exponent p, and if P has class c > 1 then P is a descendant of P/γc(P). For each group of exponent p with order less than p8 we calculate the number of descendants of order p8 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 p8, we have not been able to obtain a list of the descendants. [ABSTRACT FROM AUTHOR]