I have an acer Aspire One D255 (dual 1.5Ghz atom core and 1GB RAM running win7 Starter) with avast 5.1.864 and no issues.
I would say 1GB and windows7 really is a bit low as a lot of that will be eaten up win7. I don't have much running on this system (it is basically a backup system to my desktop) and it still regularly consumes 75% RAM, very little of that is associated to avast about 5MB for avastUI and less for avastSvc.exe, see image.