Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33791 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(15,21)(16,25)(17,20)(19,32)(22,36)(24,41)(26,38)(27,44)(28,43)(30,31)(33,34)(35,40)(37,45)(39,42);(6,8,9,13)(7,10,11,12)(14,15)(16,26)(17,29)(18,21)(19,24)(20,23)(22,28)(25,36)(27,37)(30,41)(31,35)(32,40)(33,45)(34,39)(38,43)(42,44);(8,13)(10,12)(14,16)(15,22)(17,27)(18,28)(19,33)(20,34)(21,26)(23,37)(24,42)(25,32)(29,39)(30,43)(31,44)(35,45)(36,41)(38,40);(7,11)(10,12)(14,17,18,20)(15,23,21,29)(16,27,28,34)(19,30,31,32)(22,37,26,39)(24,40,35,41)(25,33,43,44)(36,42,38,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26779);