We investigate the classification of Riemann surface topology generated by multiple arbitrary-order exceptional points of quasistationary states based on the permutation group. Our studies reveal all possible product permutations of holonomy matrices that describe a stroboscopic encircling of second-order exceptional points. The permutations turn out to be categorized into a finite number of classes according to the topological structures of the Riemann surfaces. Our results are verified by an effective non-Hermitian Hamiltonian founded on generic Jordan forms and then examined in physical systems of desymmetrized optical microcavities.