  The aim of this work is to find some exact sequences on the $c$- nilpotent
multiplier of a group $G$. We also give an upper bound for the $c$- nilpotent
multiplier of finite $p$-groups and give the explicit structure of groups whose
take the upper bound. Finally, we will get the exact structure of the $c$-
nilpotent multiplier and determine $c$-capable groups in the class of
extra-special and generalized extra-special $p$-groups. It lets us to have a
vast improvement over the last results on this topic.
