Let n be a 2-step nilpotent Lie algebra which has an inner product <,> and has an orthogonal decomposition $n=delta{oplus}varsigma$ for its center $delta$ and the orthogonal complement $varsigma;of;delta$. Then Each element Z of $delta$ defines a skew symmetric linear map $J_Z:varsigma{ ightarrow}varsigma$ given by $=$ for all $X,;Y{in}varsigma$. Let $gamma$ be a unit speed geodesic in a 2-step nilpotent Lie group H(2, 1) with its Lie algebra n(2, 1) and let its initial velocity ${gamma}$(0) be given by ${gamma}(0)=Z_0+X_0{in}delta{oplus}varsigma=n(2,;1)$ with its center component $Z_0$ nonzero. Then we showed that $gamma(0)$ is conjugate to $gamma(frac{2n{pi}}{ heta})$, where n is a nonzero intger and $-{ heta}^2$ is a nonzero eigenvalue of $J^2_{Z_0}$, along $gamma$ if and only if either $X_0$ is an eigenvector of $J^2_{Z_0}$ or $adX_0:varsigma{ ightarrow}delta$ is not surjective.