Some of the most well-studied properties of Polish groups concern the interactions between the topological and algebraic structure of such groups. Examples include the small index property and the automatic continuity property. An important method for proving them is to show that the group has an even stronger property of having ample generics. Therefore, we often want to know whether a given Polish group has a comeager conjugacy class, i.e., a generic element, a generic pair, or more generally, a generic n-tuple. After a survey on this topic, I will discuss a recent result joint with Aristotelis Panagiotopoulos, where we show that the automorphism group of the random poset does not admit a generic pair. This answers a question of Truss and Kuske-Truss.