Every group is a subgroup of an acyclic group

Statement
Every group can be realized as a subgroup of an acyclic group.

Stronger facts

 * Every group is a conjugacy-closed subgroup of an acyclic group
 * Every group is a subgroup of a binate group

Other related facts

 * Every torsion-free group is a subgroup of a torsion-free group with two conjugacy classes