on graham higman's famous porc paper

‎we investigate graham higman's paper enumerating $p$-groups‎, ‎ii‎, ‎in which he formulated his famous porc conjecture‎. ‎we are able to simplify some of the theory‎. ‎in particular‎, ‎higman's paper contains five pages of homological algebra which he uses in‎ ‎his proof that the number of solutions in a finite field to a finite set of‎ ‎monomial equations is porc‎. ‎it turns out that the homological algebra‎ ‎is just razzle dazzle‎, ‎and can all be replaced by the single observation‎ ‎that if you write the equations as the rows of a matrix then the number of‎ ‎solutions is the product of the elementary divisors in the smith normal form‎ ‎of the matrix‎. ‎we obtain the porc formulae for the number of $r$-generator groups of $p$‎ -‎class two for $rleq 6$‎. ‎in addition‎, ‎we obtain the porc formula for the‎ ‎number of $p$-class two groups of order $p^{8}$‎.