In order to achieve a high level of impermeability with a low closing force for the sealing elements of a multistage piston compressor, the overflow channel (19) in the valve piston (9) opens into at least two passages (26, 27) and the sealing plate of the overflow check valve (20) that is located in the valve piston (9) is configured as a freely guided closing membrane (28) with a limited stroke. The passages (26, 27) of the overflow channel (19) have different diameters, are arranged on a graduated circle at a radial distance from the axis of the closing membrane (28) and are completely covered by the closing membrane (28).