How dense can a *finite* set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still *transitive* on the points?

As a measure for density I use the radius of the largest spherical cap not containing any point in its interior.

For $d=1$ we can get arbitrarily dense. For $d=2$ I suppose the densest set is some orbit of the icosahedral group. Is there something known about general $d$? E.g. is there any $d>1$ for which one can get arbitrarily dense again?