In this paper, we prove some Hardy-type inequalities for the degenerate operators, $L_{p,\alpha}u\,{=}\,{\rm div}_L(|\nabla_Lu|^{p-2}\nabla_Lu)$, where $\nabla_Lu\,{=}\,(\frac{\partial u}{\partial z_1},\ldots,\frac{\partial u}{\partial z_n},|z|^\alpha \frac{\partial u}{\partial t_1},\ldots,|z|^\alpha\frac{\partial u}{\partial t_m})$. These inequalities are established for the whole space, the pseudo-ball and the external domain of the pseudo-ball. We also give a generalization of a result in [8]. Finally, a sharp inequality for $L_{\alpha}\,{=}\,L_{2,\alpha}$ is obtained.