Let n be a 2-step nilpotent Lie algebra which has an inner product <, > and has an orthogonal decomposition $n;=z;{oplus}v$ for its center z and the orthogonal complement v of z. Then Each element z of z defines a skew symmetric linear map $J_z;:;v;{longrightarrow};v$ given by <$J_zx$, y> = <z, [x, y]> for all x, $y;{in};v$. In this paper we characterize Jacobi fields and calculate all conjugate points of a simply connected 2-step nilpotent Lie group N with its Lie algebra n satisfying $J^2_z$ = <Sz, z>A for all $z;{in};z$, where S is a positive definite symmetric operator on z and A is a negative definite symmetric operator on v.