specific heat capacity

Conductivity and specific heat capacity are two independent physical quantities. Conductivity accounts for the ability of the material to [conduct] heat, whereas specific heat capacity accounts for the ability of the materual to [absorb] heat. During steady state conduction, the temperature of the conducting material doesn't change.

It is only that materials with low specific heat capacity could reach steady state condition faster than those with high specific heat capacity.

In the microscopic scale, free electrons in a material is responsible for heat conduction. This is the reason why metals, which have plenty of free electrons, are usually good heat conductors.

Specific heat capacity of solids is attributed by the lattice structure of the material. Materials with bound structure have higher specfic heat capacities.