Math Puzzle

warning: heavy abstract algebra ahead

Is it true that GL_n(Z_2) is isomorphic to the permutation group S_((2^n)-1)?

It's trivally true for n=1, and our instructor showed us a clever proof for n=2.  Given the way he proved it for n=2, I made this conjecture.  Thoughts?

GL_3(Z_2) has less than 512 elements, and S_(2^3 - 1) = S_7 has 5040 elements.