To further enhance the performance of traditional wireless multicasting, in this paper a novel multicast system is proposed based on non-orthogonal multiple access (NOMA) and device-to-device relay (D2D-relay). In the new system, all destination users are sliced into multiple tiers according to their different levels of channel states, and the multicast data are also divided correspondingly and then disseminated through power domain NOMA transmission. After executing NOMA decoding independently and locally, the destination users can share multicast data through D2D links. The joint tier slicing and power control problem is considered in this paper to minimize the overall system power consumption in the new system while satisfying that all users can successfully receive all multicast data. Extensive simulations are conducted and the superiority of the proposed new multicast system over traditional multicast system and another benchmark multicast system based on orthogonal multiple access (OMA) and D2D-relay is verified.