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) = C24 |
| direct product | C5×#12979 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1869 |
| prime radical | C5×#575 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
456 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)(8,14,22,36,21,35,52,63,23,37,54,65,28,41,58,67)(9,24,27,56,26,39,57,44,25,42,31,59,29,15,38,40)(20,47,50,68,51,34,60,62,48,55,53,69,49,61,66,64);(6,8,11,22,12,21,32,52,13,23,33,54,10,28,30,58)(7,14,17,36,18,35,45,63,19,37,46,65,16,41,43,67)(9,20,31,53,26,51,38,66,25,48,27,50,29,49,57,60)(15,34,44,64,24,55,59,68,39,61,40,62,42,47,56,69);(6,9)(7,15)(8,20)(10,29)(11,31)(12,26)(13,25)(14,34)(16,42)(17,44)(18,24)(19,39)(21,51)(22,53)(23,48)(27,33)(28,49)(30,57)(32,38)(35,55)(36,64)(37,61)(40,46)(41,47)(43,56)(45,59)(50,54)(52,66)(58,60)(62,65)(63,68)(67,69)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5967);