The dimensions of the coefficient of thermal conductivity are:

The coefficient of thermal conductivity is defined as the rate of flow of heat energy per unit area of cross-section per unit temperature gradient. The dimensions of rate of flow of heat are $\frac{{\displaystyle {\mathrm{ML}}^2{\mathrm{T}}^{-2}}}{\mathrm{T}}={\mathrm{ML}}^2{\mathrm{T}}^{-3}$.  

dimensions of area = L² 

the dimensions of temperature gradient = dimensions of temperature/dimension of length = KL^{- 1}. 

Hence, the dimensions of the coefficient of thermal conductivity are $\frac{{\mathrm{ML}}^2{\mathrm{T}}^{-3}}{{\mathrm{L}}^2 \times {\mathrm{KL}}^{-1}}={\mathrm{MLT}}^{-3}{\mathrm{K}}^{-1}.$