We investigate the properties of a frustrated spin-5/2 chain with next-nearest-neighbor two- and three-site interactions, with two questions in mind: the nature of the transition into the dimerized phase induced by the three-site interaction, and the possible presence of a critical floating phase at intermediate values of the next-nearest-neighbor interaction. We provide strong evidence that the continuous transition into the dimerized phase, which has been found to be generically in the Wess-Zumino-Witten SU(2)2S universality class up to spin S=2, is SU(2)5 only at two isolated points of the phase diagram, and that it is SU(2)3 in between, in agreement with the presence of two relevant operators allowed by symmetry for SU(2)5, and with the conservation of the parity of the level index along the renormalization flow between SU(2)k theories with different values of k. We also find that the dimerization induced by the next-nearest-neighbor interaction is a three step process, with first a small partially dimerized phase followed by a broad critical floating phase with incommensurate correlations before the fully dimerized phase is reached. Implications for the iron oxide Bi3FeMo2O12 are briefly discussed.