A Characterization of $L_2(2^f)$ in Terms of Character Zeros
  The aim of this paper is to classify the finite nonsolvable groups in which every irreducible character of even degree vanishes on at most two conjugacy classes. As a corollary, it is shown that $L_2(2^f)$ are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class. 
Guohua Qian and Wujie Shi