In this paper, device-to-device (D2D) communication is integrated into cooperative multicast to elevate the multicast transmission rate from base station to users. To this end, a new optimization problem, aiming to maximize the overall cooperative multicast rate, is formulated with two essential features 1) D2D communication is exploited to accomplish the data forwarding in the second phase of cooperative multicast, which overcomes the performance bottleneck that time resource has to be bisected for original relay forwarding in conventional cooperative multicast. 2) Social tiers are also taken into account for relay user selection, which can be viewed as a kind of incentive mechanism to enable D2D communication. To solve the problem efficiently, a two-step scheme is presented, in which relay user selection by joint leveraging channel gains and social connections, and power allocation by quantum-behaved particle swarm optimization (QPSO), are carried out sequentially. Numerical simulations show that the proposed social network-based D2D-assisted cooperative multicast scheme performs well, and the performance gap between the proposed power allocation algorithm and upper bound could be limited within 5 % when the system is not heavily loaded. The performance gain over traditional cooperative multicast can reach up to 85 %.