Funny. I'm trying to write a base50 encoder now. No good reason, just 'cuz. Can't quite figure out what to do with the half a bit. Gotta carry it forward somehow until I have a full char again but haven't come up with a good scheme.
Arithmetic coding will produce an encoding from binary to 50-ary symbols. But then again, there are much simpler ways to do that if exact base conversion is not a requirement (e.g. choose a good enough approximate solution for 2^n = 50^m and do the conversion in n-bit chunks).
You can though? You represent the encoder state in fractions of 50. In practice, it means the encoder state will not use all possible binary strings. But you don't need to trade-off in approximation size.
I am not 100% certain, but pretty convinced you can work with much smaller encoder states of only a few bytes, achieving the same compression performance of chunk approximations requiring kilobytes in size of memory.