In this paper, we prove a few facts and some cardinal properties of the space of permutation degree introduced in [6]. More precisely, we prove that if the product X n is a Lindelöf (resp. locally Lindelöf) space, then the space SP n X is also Lindelöf (resp. locally Lindelöf). We also prove that if the product X n is a weakly Lindelöf (resp. weakly locally Lindelöf) space, then the space SP n X is also weakly Lindelöf (resp. weakly locally Lindelöf). Moreover, we investigate the preservation of the network weight, π-character and local density of topological spaces by the functor of G-permutation degree. It is proved that this functor preserves the network weight, π-character and local density of infinite topological T 1-spaces.