In 1964, John Stallings established an important relationship between the low-dimensional homology of a group and its lower central series. We will establish a similar relationship between the low-dimensional homology of a group and its derived series. These results have applications in Topology. In particular, they lead to new invariants of homology equivalence of odd-dimensional manifolds and to link concordance. We will also define a solvable completion of a group that is analogous to the Malcev completion, with the role of the lower central series replaced by the derived series. We prove that the solvable completion is invariant under rational homology equivalence.