heat capacity and heat conduction

There is no connection between specific heat capacity and thermal conductivity. They are two different physical quantities.
The specific heat capacity measured the absorption of heat by a substance. The amount of heat absorbed is atttributed by the lattice (or molecular) vibration of the substance. For most solids at room temperature, the molar specific heat capacity is more or less a constant, about 25 J/mol-K
On the other hand, thermal conduction is attributed by the amount of free electrons present in the substance. This is why metals, which possess an abundance of free eelctrons, are usually good thermal conductors. &nb sp;