  The lensing properties of the Plummer model with a central point mass and
external shear are derived, including the image multiplicities, critical curves
and caustics. This provides a simple model for a flattened galaxy with a
central supermassive black hole. For the Plummer model with black hole, the
maximum number of images is 4, provided the black hole mass is less than an
upper bound which is calculated analytically. This introduces a method to
constrain black hole masses by counting images, thus applicable at cosmological
distance. With shear, the maximum number of images is 6 and we illustrate the
occurrence of an astroid caustic and two metamorphoses.
