On my 64k-ROM 512, I can run any flavor of System software, from 1.0 through the last revision prior to 6. I can even run Finder 6.0 with a System 5.x or 4.x version! But I can't get System 6 and Finder 6 to run (maybe due to lack of memory?). Despite this, Apple's website gives specs on the Mac 512 and 512ke which say these machines can both run Systems 6 and 7, which is a flat out lie.

So while you can run System 5.x, you cannot run 6.x or higher on a 512 with the original 64k ROMs.