A representation of a countable group \(\Gamma\) into a Polish group \(G\) is a homomorphism \(\Gamma \rightarrow G\). The space of all representations \(Hom (\Gamma, G)\) is a closed subspace of the Polish space \(G^{\Gamma}\) and is therefore also Polish. The authors are interested in the generic properties of representations, that hold for a comeager set of \(\pi \in Hom(\Gamma, G)\) that are moreover invariant under \(G\)-action. In many situations it turns out that the orbits are meager. Important examples of this phenomenon come from ergodic theory. The important result of the paper is Theorem 1.1. Let \(\Gamma\) be a countable group and \(G\) a Polish group. Then the set {\(\pi: \overline{\pi(\Gamma)}\) is extremely amenable} is \(G_{\delta}\) in \(Hom (\Gamma, G)\). This theorem is quite general and may be considered as a starting point for the rest of the results. Under some assumptions on \(\Gamma\), it is proven that in the first case there is a unique generic \(\overline{\pi(\Gamma)}\), and in the second it is shown that \(\overline{\pi(\Gamma)}\) is extremely amenable. It is also shown that if \(\Gamma\) is torsion-free, the centralizer of the generic \(\pi\) is as small as possible, extending a result of Chacon and Schwartzbauer from ergodic theory.