We explore the Kondo effect incorporating the localized impurity transforming under generic symmetry group G, that we call the G-Kondo effect. We derive the one-dimensional effective model coupled with the impurity, and studied the thermodynamic properties based on the boundary conformal field theory approach. We in particular study the impurity entropy at the conformal fixed point, and the low temperature behavior of the specific heat and the susceptibility. We also consider the Wilson ratio based on these results, and mention the case with exceptional group symmetry.