Abstract This paper is a contribution to the study of the automorphism groups of finite linear spaces. Let G act as a line-transitive and point-primitive automorphism group of a non-trivial linear space S with v points and lines of size k . Set k 2 = ( k , v − 1 ) . Assume q = p a for some prime p and positive integer a . If q ⩾ 3 ( k 2 k − k 2 + 1 ) a , then the socle of G is not equal to E 6 2 ( q ) .