Universality is a key hypothesis in mechanistic interpretability -- that different models learn similar features and circuits when trained on similar tasks. In this work, we study the universality hypothesis by examining how small networks learn to implement group compositions. We present a novel algorithm by which neural networks may implement composition for any finite group via mathematical representation theory. We then show that these networks consistently learn this algorithm by reverse engineering model logits and weights, and confirm our understanding using ablations. By studying networks trained on various groups and architectures, we find mixed evidence for universality: using our algorithm, we can completely characterize the family of circuits and features that networks learn on this task, but for a given network the precise circuits learned -- as well as the order they develop -- are arbitrary.