The different pinning strengths of the flux line lattice in the peak effect (PE) region of a polycrystalline sample of CeRu$_2$ with a superconducting transition temperature {$T_c = 6.1$ K} have been probed by means of magnetization measurements with a SQUID magnetometer as the temperature $T$ and the magnetic field $H$ are varied. Magnetic relaxation measurements were used to monitor the flux line dynamics in the PE region. For {$T < 4.5$ K} and $H < H_P$, where $H_P$ is the field where the magnetization reaches a maximum in the PE region, the relaxation rate was found to be significantly larger in the descending-field branch of the PE than it is in other sections of the PE region. For {$T geq 4.5$ K}, the relaxation rate in the entire PE region is so large that the magnetization reached a stable (equilibrium) value within {$10^4$ s}. This experimentally determined stable state appears as an increase of the magnetization in the PE region and has a dome shape superimposed on a linear interpolation through the PE region. It was also found that the PE in CeRu$_2$ can be suppressed by rapid thermal cycling of the sample between {10 K} and {300 K} four times. The reversible magnetization after the PE has been suppressed coincides with the linear interpolation through the PE region, in contrast to the behavior of the equilibrium magnetization when the PE is present. PACS number: 74.25.Qt, 74.70.Ad