I would like to know examples of $G/K$ where $G$ is a locally compact group and $K$ is a compact subgroup of $G$.
I know about Riemannian symmetric spaces of Euclidean, compact and non-compact type. They can be realized as $G/K$. I am wondering what are other prominent spaces (discrete and continuous) which can be written as $G/K$, where $G$ can be unimodular or non-unimodular. Perhaps homogeneous tree is an example outside the set of Riemannian symmetric spaces.