  The isothermal magnetoresistance (MR) with magnetic field (H) parallel to and
perpendicular to ab plane is systematically studied on the single crystal
Na$_{0.52}$CoO$_2$ with charge ordering at $\sim 50$ K and an in-plane
ferromagnetism below 25 K. The isothermal MR behavior with H $\parallel$ ab
plane and H $\perp$ ab plane is quite different. When H $\parallel$ ab plane,
the MR is always negative and the in-plane ferromagnetic behavior is enhanced.
While the MR with H $\perp$ ab plane changes from negative to positive with
decreasing temperature or increasing H, and the in-plane ferromagnetic behavior
is suppressed. A striking feature is that the MR with H $\perp$ ab plane shows
a hysteresis behavior below 25 K, which is absent for the case of H $\parallel$
ab plane. These results provide strong evidence for a spin-flop transition of
small moments of Co$^{3.5-\delta}$ sites induced by H $\perp$ ab plane, leading
to a metamagnetic transition for small moments of Co$^{3.5-\delta}$ sites.
These complex magnetism suggests an unconventional superconductivity in
Na$_x$CoO$_2$ system because the Na$_x$CoO$_2$ around x=0.5 is considered to be
the parent compound of superconductivity.
