There are no face-transitive polyhedra with five sides and there are two distinct topological types.
With regular polygon faces, the two topological forms are the square pyramid and triangular prism.
An irregular pentahedron can be a non-convex solid: Consider a non-convex (planar) quadrilateral (such as a dart) as the base of the solid, and any point not in the base plane as the apex.
There is a third topological polyhedral figure with 5 faces, degenerate as a polyhedron: it exists as a spherical tiling of digon faces, called a pentagonal hosohedron with Schläfli symbol {2,5}.
It has 2 (antipodal point) vertices, 5 edges, and 5 digonal faces.