Constrained by the limited resource, high-latency and high bit error rate, the existing group key management schemes for the space network are inefficient. To solve this problem, we proposed a centralized and identity-based key management scheme by using McEliece public-key cryptosystem (PKC). In this scheme, the node identity is used as the parameter to generate the public key. Therefore, the authentication can be embedded into the verification of the public key without needing the PKI. The group key is distributed with the protection of the public key so that the group key management scheme can be implemented safely. Furthermore, the McEliece public-key cryptosystem can resist the quantum attack and provide error correction capacity. It improves the efficiency of the group key distribution over the noisy channel. The proposed key management scheme is simulated on OPNET. The security of public-key generation, forward secrecy, backward secrecy and performance are analyzed. The results show that our scheme can provide confidentiality, integrity, authentication, non-repudiation, failure tolerance and error correction. In addition, the computation overhead and rounds of interaction are lower than former work.