  We study the automorphism group of a compact 7-manifold $M$ endowed with a
closed non-parallel G$_2$-structure, showing that its identity component is
abelian with dimension bounded by min$\{6,b_2(M)\}$. This implies the
non-existence of compact homogeneous manifolds endowed with an invariant closed
non-parallel G$_2$-structure. We also discuss some relevant examples.
