Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C23×C5 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 73 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 73 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,22)(13,18)(14,30)(15,35)(16,19)(17,24)(20,33)(21,32)(23,47)(25,54)(26,48)(27,42)(28,57)(29,60)(31,45)(34,46)(36,44)(37,50)(38,53)(39,62)(40,59)(41,56)(43,52)(49,69)(51,72)(55,68)(58,67)(61,65)(63,73)(64,70)(66,71);(10,12,16,26)(11,17,19,38)(13,23,27,49)(14,31,34,62)(15,36,32,25)(18,37,42,64)(20,44,43,54)(21,45,35,39)(22,46,48,30)(24,52,53,33)(28,58,55,73)(29,61,56,51)(40,65,66,72)(41,67,60,63)(47,68,69,57)(50,71,70,59);(6,8,9,7)(10,13,16,27)(11,18,19,42)(12,23,26,49)(14,28,34,55)(15,29,32,56)(17,37,38,64)(20,40,43,66)(21,41,35,60)(22,47,48,69)(24,50,53,70)(25,51,36,61)(30,57,46,68)(31,58,62,73)(33,59,52,71)(39,63,45,67)(44,65,54,72);(11,19)(13,27)(17,38)(20,43)(21,35)(22,48)(23,49)(28,55)(29,56)(30,46)(39,45)(44,54)(50,70)(51,61)(58,73)(59,71)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26446);