Let G be an algebraic group and let X be a smooth
integral scheme over a field k. In this paper we construct
homology-type groups Hi(X,G) by considering cycles
in the simplicial scheme BG x X (an idea suggested
by Andrei Suslin). We discuss the basic properties of these groups
and construct a spectral sequence, beginning with the groups
Hi(\Deltaj,G), which converges to the etale
cohomology of the simplicial group BG. These groups are
therefore connected with the study of Friedlander's generalized
isomorphism conjecture. We also compute some examples, focusing in
particular on the case X=Spec(k). In the case where
k is the real numbers, there is a connection between the
groups Hi and the Z/2-equivariant cohomology of the
classifying space of the discrete group G(R).