The gauge group is the smooth coordinate transformations (diffeomorphisms, technically), which includes the Poincare group as a subgroup.
Penrose has a brief discussion of how Noether's theorem relates (or doesn't relate) to this on p. 489 of The Road to Reality: "For example, it is not at all a clear-cut matter to apply these ideas to obtain energy-momentum conservation in general relativity, and strictly speaking, the method does not work in this case. The apparent gravitational analogue [of EM gauge symmetry] is 'invariance under general coordinate transformations' [...] but the Noether theorem does not work in this situation, giving something of the nature '0=0'."
I don't understand the technical details of N's theorem well enough to know what exactly it is that fails in this case. I believe that what she published in 1918 is actually a very restricted version of the theorem, whereas what physicists refer to today as "Noether's theorem" is actually a loosely defined set of generalizations of the 1918 version.
There are, however, some pretty simple and fundamental reasons why it *can't* apply to GR. If it did apply to GR, then the conserved quantity it gave would have to be the energy-momentum four-vector. But Gauss's theorem fails in a curved spacetime when the conserved quantity isn't a scalar, basically because any attempt to define the total flux through a surface is subject to the ambiguity introduced by having to parallel-transport the flux (which is a vector in this case) from one part of the surface to another.