Chlorine is an element in the third period of the Periodic Table. It is a non-metallic element as are phosphorus and sulfur. Generally speaking, if an element is to have the ability to conduct electricity is must be metallic. The element chlorine cannot conduct electricity because its outer electrons do not have the ability to move, to work as a conductor, because they have strong covalent bonds. The definition of a covalent bond is: a type of chemical bond involving the sharing of electrons between atoms in a molecule, especially the sharing of a pair of electrons by two adjacent atoms. These are strong bonds that make elements like chlorine, phosphorus, and sulfur unable to act as conductors.
The relative conductivity of chlorine is equal to zero.