Countable dense homogeneity and topological groups

Building on results of Medvedev, we construct a ZFC example of a non-Polish topological group that is countable dense homogeneous. Our example is a dense subgroup of Z ω of size b that is a λ -set. We also conjecture that every countable dense homogeneous Baire topological group with no isolated points contains a copy of the Cantor set, and give a proof in a very special case.