Heating a solid material with laser-accelerated fast electrons is a particularly useful method for generating a plane powerful shock wave with a pressure of several hundred or even thousands of Mbar in the laboratory. Behind the front of such a powerful shock wave, dense plasma is heated to a temperature of several keV. Then, a high rate of radiation energy loss occurs even in low-Z plasmas. In this paper, the strong compression of matter due to radiation cooling in a Gbar shock wave driven by fast electrons is studied using both computational and theoretical approaches. It is shown that the effect of radiation cooling leads to compression of matter in the peripheral region of the shock wave to a density several times greater than the density at its front. Heating a solid material by a petawatt flux of laser-accelerated fast electrons offers the opportunity to surpass the gigabar pressure level of plane shock waves generated by the impact of laser-accelerated pellets. Higher pressures of about 100 Gbar can be achieved under laboratory conditions only when a spherical target is imploded under the action of a terawatt laser pulse.