First, what is that Shanks and Sloane quotes you give above? Second, what is ? Third, I think you're making a big mix of several very different things and I'm no sure how you're dealing with this stuff if, as you say, you're "new at group theory"...!?!
You take the group of units in the ring of (integer) residues modulo 32 (and somehow I feel you'd rather have to deal with the multiplicative group of the FIELD with elements...) , but whatever: we have a nice theorem in group theory that says that the units group in is cyclic of order 6, but NOT all its non-unit elements are generators since this is not a group of order a prime: we have generators here: , and 4 cannot be a generator since and its power is not coprime with 6...For example, as 5 is coprime with 6 so 5 is a generator.
I think it'd be a good idea if you'd explain what is this about, explain the background and the symbols. Perhaps that way there's a chance somebody will be able to explain you something.