Suppose we have a convex polyhedron in $\Bbb R^n$ symmetric then we know there is an unique ellipsoid of maximum volume called John's ellipsoid inscribed in it. 

1. What symmetry does the convex polyhedron has to have for its barycenter (arithmetic mean of vertices) to agree with the center of John's ellipsoid?

2. Also what is the minimal number of $2$-faces of the polytope do we have to know to obtain the ellipsoid exactly when the polyhedra has such symmetry?