The (r, R) properties of regular point systems (RPS) of the general positions are discussed. The theorem relating the R characteristics of the RPS of the general position in space group G to the diameter of the fundamental domain F_G of this group is proved. Using this theorem, the upper bounds of the R characteristics of the RPS of the general positions in the space groups of cubic system are determined as functions of the lattice constant a. Considering the space groups Pn3̄ n and Fd3̄ c, it was shown that, for asymmorphic and hemisymmorphic groups, these bounds can be reduced upon the detailed analysis of the geometries of the corresponding fundamental domains.