In this paper, we study the abelian complexity $ρ_n^{ab}(\mathbf{t}^{(k)})$ of generalized Thue-Morse sequences $\mathbf{t}^{(k)}$. We obtain the exact value of $ρ_n^{ab}(\mathbf{t}^{(k)})$ for every integer $n\geq k$. Consequently, $ρ_n^{ab}(\mathbf{t}^{(k)})$ is ultimately periodic with the period $k$. Moreover, we show that the abelian complexities of a class of infinite sequences are $k$-automatic.