To every topological group G we can associate its Samuel compactification (S(G), 1). This is the largest point-transitive G-flow according to a suitable universal property. Using the universal property, we can endow S(G) with the structure of a compact left-topological semigroup. While the algebraic properties of S(G) are an active area of research for G a countable discrete group, less attention has been paid to other topological groups. In this talk, we will discuss a method of characterizing S(G) when G is an automorphism group of a countable structure. We will then take a closer look at the case G = S_∞ and answer several questions about the algebraic structure of S(G). This is joint work with Dana Bartošová.