All spaces are assumed to be separable and metrizable. A space X is countable dense homogeneous (CDH) if all countable dense subsets of X can be mapped onto each other by homeomorphisms of X. The fundamental theorem of countable dense homogeneity states that every “sufficiently homogeneous” Polish space is CDH. This result motivated a long-standing search for examples of non-Polish CDH spaces. We contribute to this line of research by exhibiting a non-Polish CDH topological group. This is joint work with Claudio Agostini and Lyubomyr Zdomskyy.