An electrical machine has a stator with a stator body supporting an electrical stator and a rotor. The rotor is supported by bearing including a radial bearing section forming a radial gas bearing and an axial bearing section forming an axial gas bearing, the stator side parts of these bearing sections being a stator side radial bearing part and a stator side axial bearing part that are rigidly connected to one another and together form a stator bearing structure. The stator side radial bearing part is a bushing, and is radially surrounded by a cooling body. The bushing is connected to the cooling body by an elastic support comprising a thermally conducting filler.