  We study the problem of gravitational lensing by an isothermal elliptical
density galaxy in the presence of a tidal perturbation. When the perturbation
is fairly strong and oriented near the galaxy's minor axis, the lens can
produce image configurations with six or even eight highly magnified images
lying approximately on a circle. We classify the caustic structures in the
model and identify the range of models that can produce such lenses. Sextuple
and octuple lenses are likely to be rare because they require special lens
configurations, but a full calculation of the likelihood will have to include
both the existence of lenses with multiple lens galaxies and the strong
magnification bias that affects sextuple and octuple lenses. At optical
wavelengths these lenses would probably appear as partial or complete Einstein
rings, but at radio wavelengths the individual images could probably be
resolved.
