<p>The purpose of this paper is to give a complete description of some basic Riemannian objects on the free step-two Carnot group <i>N</i>_3,2. The author first finds out all the conjugate points along any geodesic starting from o by means of calculating the Jacobian determinant of the Riemannian exponential map. Next, the author determines the cut locus and the explicit expression of the squared distance <i>d</i>²(<i>g</i>). Moreover, the author obtains the optimal syntheses from <i>o</i> and proves that the first conjugate point coincides with the cut point.</p>