On Graham Higman's famous PORC paper

We investigate Graham Higman''s paper Enumerating p-groups, II, in whichhe formulated his famous PORC conjecture. We look at the possibilities for turning his theory into a practical algorithm for computing the number of p-class two groups of order pn for small n. We obtain the PORC formulae for the number of r-generator groups of p-class two for r  6. In addition, we obtainthe PORC formula for the number of p-class two groups of order p8.One of the ideas used in implementing Higman''s theory has led to a signi -cant speed up in Eamonn O''Brien''s ClassTwo function in Magma. In addition,we are able to simplify some of the theory. In particular, Higman''s paper con-tains ve pages of homological algebra which he uses in his proof that the number of solutions in a nite eld to a nite 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.