The electronic structure and electronic conductance of C20 molecule bridged between two Au electrodes are studied by the first principles based on the density functional theory and the nonequilibrium Greens function. The three transmission systems based on C20 molecules are modeled，and the transmission spectra and the molecular orbital distributions are obtained. The reasons why the electronic structure and the electronic transmission are formed are discussed. The results show that the electrons are transported through the C20 bridge on the molecular shell. When one K or one Si atom is inlaid inside C20，the conductivities indicate that the electrons can transfer partly inside the C20 molecule，but most electrons ave transported on the C20 fullerene shell.