We present a comparative study on the structural properties of the coinage metal icosamers Cu(20)(+/-), Ag(20)(+/-), and Au(20)(+/-). Using trapped ion electron diffraction measurements in combination with density functional structure calculations we find distinct structural differences depending on the cluster material and the charge state: Cu(20)(-), Cu(20)(+), as well as Ag(20)(+) prefer icosahedral structures. Ag(20)(-) adopts a rearranged, distorted icosahedral structure. While Au(20)(-) is tetrahedral, Au(20)(+) cannot be described satisfyingly by a single isomer alone. Here a mixture of tetrahedral and distorted icosahedral structures is suggested. The influence of material and charge on the structural properties of the coinage metal icosamers is discussed.