This investigation was originally motivated by the problem of determining the maximum number of variables which can be accommodated in a $2_v^{k-p}$ design in 256 runs and of constructing such a "saturated" design. This problem is solved through the application of an algorithm given by the authors in a previous paper (Draper and Mitchell (1967)) to the particular case $R = 5, q = k - p = 8$. To obtain the solution, the complete set of even 512-run designs of resolution $\geqq 6$ and the complete set of 256-run designs of resolution $\geqq 5$ are constructed. Tables are given which immediately provide generating relations for all of these designs, "optimally" blocked.