Gait patterns of insects and animals are generated by Central Pattern Generators (CPGs) in their nervous systems. The dynamics of the CPG are often modeled mathematically using a system of coupled neural oscillators, which can generate various animal-like gait patterns. It is expected that the application of the CPG model to control or design of walking robots can realize stable and adaptive legged locomotion. In this study, we analyze the CPG model proposed by Golubitsky, which consists of symmetrically coupled FitzHugh-Nagumo oscillators. The gait patterns correspond to stable limit cycles of the model. To assess the stability of the gait patterns of the CPG model, we calculate the Lyapunov exponents and covariant Lyapunov vectors of the limit cycle corresponding to each gait pattern, which characterize linear response property of infinitesimal perturbations added to the limit-cycle orbit; the Lyapunov exponents measure the exponential growth rates of the perturbations and the covariant Lyapunov vectors characterize their corresponding directions. In particular, We focus on the second Lyapunov exponent and the associated covariant Lyapunov vector in order to identify the perturbation patterns with the longest decay time, i.e., the most influential external disturbances on the gait. We also show that the covariant Lyapunov vectors reflect the symmetry of the CPG model.