Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
476 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)(7,10)(8,12)(14,15,19,24)(16,20,25,33)(17,21,30,37)(18,31,29,23)(22,38,36,28)(26,32,39,43)(27,42,40,35)(34,45,44,41);(6,7)(8,11)(9,10)(12,13)(14,16)(15,20)(17,26)(18,27)(19,25)(21,32)(22,34)(23,35)(24,33)(28,41)(29,40)(30,39)(31,42)(36,44)(37,43)(38,45);(7,10)(11,13)(15,21)(17,28)(18,29)(20,32)(22,23)(24,37)(26,41)(27,40)(30,38)(31,36)(33,43)(34,35)(39,45)(42,44);(6,8,9,12)(7,11,10,13)(16,25)(20,33)(26,39)(27,40)(32,43)(34,44)(35,42)(41,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26812);