Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C25 |
| direct product | C5×#12685 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1779 |
| prime radical | C5×#585 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
890 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)(6,7,11,17,12,18,32,45,13,19,33,46,10,16,30,43)(8,20,24,48,22,36,54,44,23,41,31,59,28,14,34,37)(9,15,26,39,25,38,55,64,27,40,57,66,29,42,58,67)(21,49,52,69,53,35,61,63,51,47,56,68,50,62,60,65);(6,8)(7,14)(9,21)(10,28)(11,31)(12,22)(13,23)(15,35)(16,41)(17,44)(18,20)(19,36)(24,33)(25,53)(26,56)(27,51)(29,50)(30,54)(32,34)(37,46)(38,47)(39,65)(40,62)(42,49)(43,48)(45,59)(52,57)(55,60)(58,61)(63,66)(64,69)(67,68);(6,9)(7,15)(8,21)(10,29)(11,26)(12,25)(13,27)(14,35)(16,42)(17,39)(18,38)(19,40)(20,47)(22,53)(23,51)(24,52)(28,50)(30,58)(31,56)(32,55)(33,57)(34,60)(36,62)(37,63)(41,49)(43,67)(44,65)(45,64)(46,66)(48,68)(54,61)(59,69)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5673);