Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12830 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #418 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1804 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 69 points:
(1,2,3,4,5)(6,7,11,18,13,16,12,35)(8,19,22,40,23,14,34,39)(9,24,27,56,25,57,36,46)(10,28,31,44,33,60,32,17)(15,30,41,58,43,64,42,26)(20,48,51,69,49,65,52,67)(21,53,55,45,29,61,63,38)(37,54,66,50,47,62,68,59);(6,8,12,22,13,23,11,34)(7,14,35,39,16,19,18,40)(9,20,36,51,25,49,27,52)(10,29,32,63,33,21,31,55)(15,37,42,66,43,47,41,68)(17,45,60,61,44,38,28,53)(24,48,46,69,57,65,56,67)(26,59,64,54,58,50,30,62);(6,9)(7,15)(8,20)(10,30)(11,27)(12,36)(13,25)(14,37)(16,43)(17,46)(18,41)(19,47)(21,54)(22,51)(23,49)(24,28)(26,32)(29,62)(31,58)(33,64)(34,52)(35,42)(38,67)(39,66)(40,68)(44,56)(45,69)(48,53)(50,55)(57,60)(59,63)(61,65)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5818);