Ab initio calculations of magnetic moments for icosahedral clusters contained in crystal structures Al10Mn3, Al5Co2, Al17Mn4 (Al13Cr4Si4-type) fulfilled in the framework of Density Functional Theory. The AlMn cluster having the trigonal D3h symmetry with the triangle of Mn ions in the interior has the moment being equal to three magnetic moments of a single manganese ion (4.4 μB), the moment of the tetrahedral Td cluster with the Mn tetrahedron in the interior is equal approximately to twelve magnetic moments of the single manganese ion (15.5 μB). The magnetic moment of icosahedral Al-Co clusters having the same configuration is equal to zero. The magnetic moments of the rod assembled from the icosahedral clusters with the sequence Td D3h - Td was found to be 20.5 μB. This value permits to explain the giant magnetic moment of icosahedral and decagonal Al-Mn quasicrystals and gives the indirect evidence to the hierarchical model of the quasicrystals structure proposed by the authors recently. An arrangement of magnetic moment carriers in the interior of the aluminum shell of icosahedral clusters permits to suggest the interaction between contacting manganese ions as the main origin of the giant magnetic moment of the Al-Mn quasicrystals.