A cylinder for processing sheets of printing material includes a clamping channel having a central region and edge regions. A tool plate punches and/or scores and/or perforates the sheets. A support plate carries the tool plate. A first clamping system clamps the support plate and has a first holding device for holding a front end of the support plate and a second holding device for holding a rear end of the support plate. The first holding device and the second holding device are disposed in the edge regions. A second clamping system clamps the tool plate and has a third holding device for holding a front end of the tool plate and a fourth holding device for holding a rear end of the tool plate. The third holding device and the fourth holding device are disposed in the central region. A printing press having the cylinder, is also provided.