Assume a group $G$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $H$ in such a way that the orbits are closed and the quotient $H/G$ is isometric to a finite dimensional Riemannian manifold.

Edit after YCor comment: Is there a natural way to equip $G$ with a topology so that it is an infinite dimensional Lie Group with a Lie Algebra?

1more comment