  We determine the most general form of the antisymmetric $H$-field tensor
derived from a purely time-dependent potential that is admitted by all possible
spatially homogeneous cosmological models in 3+1-dimensional low-energy bosonic
string theory. The maximum number of components of the $H$ field that are left
arbitrary is found for each homogeneous cosmology defined by the Bianchi group
classification. The relative generality of these string cosmologies is found by
counting the number of independent pieces of Cauchy data needed to specify the
general solution of Einstein's equations. The hierarchy of generality differs
significantly from that characteristic of vacuum and perfect-fluid cosmologies.
The degree of generality of homogeneous string cosmologies is compared to that
of the generic inhomogenous solutions of the string field equations.
