Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C22×C42 |
| direct product | C5×#29812 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,22,26,23,27,35,41)(20,24,30,38,29,37,45,48)(21,31,32,39,33,40,34,25)(28,42,44,47,43,49,46,36);(6,7)(8,9)(11,14)(15,17)(18,20)(19,24)(21,28)(22,30)(23,29)(25,36)(26,38)(27,37)(31,42)(32,44)(33,43)(34,46)(35,45)(39,47)(40,49)(41,48);(7,9)(12,16)(15,17)(18,21,23,33)(19,25,27,39)(20,28,29,43)(22,34,35,32)(24,36,37,47)(26,40,41,31)(30,46,45,44)(38,49,48,42);(10,12,13,16)(11,15,14,17)(20,29)(24,37)(28,43)(30,45)(36,47)(38,48)(42,49)(44,46)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22800);