In recent years, GaN doped with Gd (GaN:Gd) has attracted much attention due to its potential applications in spintronic devices since the high temperature ferromagnetism and the colossal magnetic moment were observed in GaN:Gd. However, the microscopic nature of ferromagnetism in GaN:Gd still is controversial. We investigate the crystal parameters, magnetic moment, formation energies, and electronic structures of the defect complexes formed by Gd and native Ga (or N) vacancies in GaN by using the first-principles method based on the density functional theory. The calculated results show that the energy band gap of GaN:Gd becomes indirect and its width becomes small compared with that of GaN. The lattice constants of GaN:Gd expand due to the larger ionic radius of Gd than that of Ga atom, while they shrink when the Gd atom and Ga vacancies coexist. In the case of the isolated Gd dopant, the Gd-4f electrons lead to a magnetic moment of about 7 B in GaN:Gd. For the defect complex, one Ga vacancy can introduce a magnetic moment of about 2 B, while N vacancy has little effect on the total magnetic moment. In addition, when we focus on the defect complex composed of Gd and five neighboring Ga vacancies, we find that the magnetic moment of per Gd atom and the total magnetic moment depend strongly on the concentration and position of Ga vacancies. When the Ga vacancies are distributed loosely near the Gd atom, the magnetic moment of Gd atom increases slightly, while for the closely-distributed Ga vacancies the Gd magnetic moment can be increased by 2 B. We infer that the interactions among Ga vacancies result in the large magnetic moment of Gd atom. It is also found that the formation energy is very small when the Ga vacancies are distributed thickly around the Gd atom in GaN:Gd. Our results are in qualitative agreement with the results from other studies (Thiess A et al. 2012 Phys. Rev. B 86 180401; Thiess A et al. 2015 Phys. Rev. B 92 104418), where Ga vacancies were proposed to tend to cluster in GaN:Gd and induce the large magnetic moment of Gd. Moreover, the effect of distance between the Gd atom and Ga vacancies on the Gd magnetic moment is also discussed. It is found that the Gd magnetic moment is relatively large when Ga vacancies are close to the Gd atoms.