Many non-locally compact second countable groups admit a comeagre conjugacy class. For example, this is the case for S ∞ S_{\infty } , A u t ( Q , > ) Aut(\mathbb {Q},>) , and, less trivially, A u t ( R ) Aut(\mathcal {R}) for R \mathcal {R} the random graph. A. Kechris and C. Rosendal ask if a non-trivial locally compact second countable group can admit a comeagre conjugacy class. We answer the question in the negative via an analysis of locally compact second countable groups with topological conditions on a conjugacy class.