Of course, it all starts at the top, with Joseph Stalin. He's not an actual character in the book (except when we see him in Sasha's imagination and in all those statues of assorted sizes), but it's safe to say that he and the entire political system are the major antagonists because of the havoc they wreak on Sasha's life.
There are also a boatload of other assorted meanies and baddies that work in Stalin's name. So, the Senior Lieutenant who arrests Sasha's dad, the State Security officers who drive around in their spooky black cars and kick people's doors in at midnight, and Nina Petrovna (Sasha's teacher who is an A-level schemer and master of psychological abuse) should all be lumped into the antagonist category. But again, they fall under the larger umbrella of the Soviet system, which is the real enemy here.