The full shift is the topological dynamical system of bi-infinite sequences over a finite alphabet under the shift map. Its automorphism group, the group of (reversible) cellular automata, has a rich family of finitely-generated subgroups, including all finite, f.g. abelian and f.g. free groups, and several groups with undecidable torsion problem. Many closure properties are also known for this family, such as closure under direct products and free products. The group itself is not finitely-generated. Nevertheless, we show that there is a finitely-generated subgroup that is universal, in the sense that it contains an embedded copy of every other finitely-generated subgroup.
