From the GM's handbook: "During character creation each player draws a card and keeps it with them..."
So, I totally see why you'd think that means it's random. However, this IS part of a massive chapter on Secret Societies, providing details about the societies that only the GM knows, so I think it's pretty clear that's not supposed to mean you don't know about it. Or, if you prefer...
THE FIVE STAGES OF READING A RULE YOU DON'T LIKE
DENIAL - I know it says that, but it's obviously not what they meant. Besides, the player's handbook states it's coming from the GM.
ANGER - THAT'S JUST WRONG. IT'S STUPID. IF THE WRITER WAS HERE AND THEY SAID "YEAH, IT'S SUPPOSED TO BE RANDOM", I'D YELL AT THEM SO HARD I'D MAKE THEM CRY. STUPID, STUPID RULE.
BARGAINING - I mean, technically it just says they draw it, maybe you could go ahead and look at it because you're the GM? I mean, you should have a copy of their character sheet and everything, wouldn't you want to have their mutant power and secret society information?
DEPRESSION - I've been playing since the WEG days, and I had so many of the old Mongoose XP/25th books, and I've told everyone that they're the best game publishers in the world, and I'm just so upset by this one little rule that I don't know why I even bother playing any more...
ACCEPTANCE - Of course, on page 104, it also says "THE RULES ARE IN YOUR FAVOR" so, if you want to play that as written, go ahead and hope they take advantage.