As others have said, different relationships have different boundaries.
However, I would be so upset if my boyfriend did this to me. I don't mind strip clubs and going out with your mates to them, but I draw the line at private dances for the very reason you've just mentioned - it's too tempting. And the idea of kissing her nipples or touching her bum...that makes my skin crawl.
I'm sorry, but you said that you told your girlfriend that you were going first (which was good, let me just say), and that you might end up being persuaded into things by your friends. You said that she said okay but was visibly saddened by this. Surely that should have been an indication that she wasn't really okay with this, and so you should have gone out of your way to not make her insecure, aka, not taking liberties with her permission by getting a private dance. Also, you're a big boy - can't you say no to your friends? If they were your friends they wouldn't be trying to put you in a position which might (in a weaker case) break your relationship.
You didn't cheat, but I certainly don't think you were right, and if I was your girlfriend I would feel like my trust had been betrayed, and I would be watching you a bit more closely now. I let you go to a strip club, and you took advantage of my trust.
Having said that, the fact that you're feeling guilty for it is good, and it's clear that you don't want to do it again. I'd be really on my guard if I were you though, as clearly you can be tempted if the opportunity arises, and if you love this girl, you'll understand that doing that will hurt her.