I have student visa with my dependent for UK, can my dependent work full time?

Is your dependent British or from a European Union country? If he or she is then the answer is yes. If not, then the answer is a qualified yes.

He or she would need to apply for a work permit in order to be able to work (that's true for part-time as well as full-time work).

Yes, there IS some "cash in hand" work available but it's generally poorly paid, lousy types of job and it's also illegal!

Best to try to get a work permit first. Apply to the British Consul and they'll give you a detailed answer but that's the basic principles of it.