Homogeneous space of the unitary group U(n)

Question

Consider the unitary group $U(2^n)$ and let $G$ be the subgroup isomorphic to $U(2^{n-1})$ embedded as: \begin{pmatrix} U & 0\\ 0 & U \end{pmatrix} where $U \in U(2^{n-1})$ (the same matrix in both blocks). How to describe the homogeneous space $U(2^n) / G$? Do we get a Stiefel manifold as in the case of embedding: \begin{pmatrix} Id & 0\\ 0 & U \end{pmatrix}