  We classify the $5$-dimensional homogeneous geometries in the sense of
Thurston. The present paper (part 2 of 3) classifies those in which the linear
isotropy representation is either irreducible or trivial. The $5$-dimensional
geometries with irreducible isotropy are the irreducible Riemannian symmetric
spaces, while those with trivial isotropy are simply-connected solvable Lie
groups of the form $\mathbb{R}^3 \rtimes \mathbb{R}^2$ or $N \rtimes
\mathbb{R}$ where $N$ is nilpotent.
