The author considers singular square \(n\times n\) matrices A which are of index one (rank(A \(2)=rank(A)=n-m)\). The value m is assumed much smaller than n. The author also considers block systems of k copies of A along the diagonal \((k\ll n)\), the identity on the subdiagonal, and zero elsewhere. The interest is in computing the group inverse of A, denoted \(A^{\#}\), which is the Drazin generalized inverse for the case of index one. The group inverse is more convenient in certain applications than the Moore-Penrose pseudo-inverse. An algorithm of \textit{J. H. Wilkinson} [Res. Notes Math. 66, 82-99 (1982; Zbl 0491.65025)] can be used to compute \(A^{\#}\), with an asymptotic operation count of 2n 3 multiply-adds. This paper contains a proof that certain intermediate (n-m)\(\times (n-m)\) matrices can be replaced by \(m\times m\) matrices, thereby reducing the asymptotic operation count to 2n 3/3 multiply-adds.