I believe the Local Group is gravitationally bound, so there is a common center about which its mass must collectively orbit. However, the group is only about 100 times the diameter of the Milky Way, so galaxies are much larger in comparison to the spaces between them than stars are, so they interact in more complex ways than stars typically do - ripping each other apart, for example.
All that means that I think you have a mass distribution that isn't really well described as a fluid (like a galaxy) or a collection of point masses orbiting one (like a solar system). So I would suspect that galaxy motions aren't pure ellipses, because they interact a lot (and there are definitely satellite galaxies of the Milky Way). But I am not certain.