Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12947 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1862 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
380 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)(6,7,12,19,14,20,13,35)(8,21,25,51,26,52,36,39)(9,27,30,48,31,63,37,43)(10,18,32,47,33,49,11,17)(15,24,40,59,42,61,41,23)(16,28,44,65,46,34,45,29)(22,53,56,67,58,68,57,38)(50,60,66,64,69,54,62,55);(6,8,12,25,14,26,13,36)(7,15,19,40,20,42,35,41)(9,22,30,56,31,58,37,57)(10,24,32,59,33,61,11,23)(16,38,44,53,46,67,45,68)(17,39,18,21,47,51,49,52)(27,62,48,50,63,66,43,69)(28,64,65,54,34,55,29,60);(6,9)(7,16)(8,22)(10,29)(11,34)(12,30)(13,37)(14,31)(15,38)(17,27)(18,48)(19,44)(20,46)(21,50)(23,55)(24,60)(25,56)(26,58)(28,32)(33,65)(35,45)(36,57)(39,62)(40,53)(41,68)(42,67)(43,49)(47,63)(51,66)(52,69)(54,61)(59,64)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5935);