Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 69 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 69 points:
(1,2,3,4,5)(6,7,13,19)(8,20,25,38)(9,26,30,45)(10,31,34,43)(11,18,12,17)(14,24,39,23)(15,35,42,29)(16,33,44,32)(21,40,27,36)(22,50,53,37)(28,58,60,41)(46,54,55,49)(47,52,63,51)(48,67,57,66)(56,61,65,59)(62,68,64,69);(6,8,12,24,13,25,11,23)(7,14,18,38,19,39,17,20)(9,27,29,54,30,21,35,49)(10,22,33,52,34,53,32,51)(15,40,26,55,42,36,45,46)(16,37,43,63,44,50,31,47)(28,57,59,69,60,48,61,68)(41,64,56,67,58,62,65,66);(6,9)(7,15)(8,21)(10,28)(11,29)(12,35)(13,30)(14,36)(16,41)(17,26)(18,45)(19,42)(20,46)(22,48)(23,49)(24,54)(25,27)(31,56)(32,59)(33,61)(34,60)(37,62)(38,55)(39,40)(43,65)(44,58)(47,66)(50,64)(51,68)(52,69)(53,57)(63,67);(6,10,12,33,13,34,11,32)(7,16,18,43,19,44,17,31)(8,22,24,52,25,53,23,51)(9,28,35,61,30,60,29,59)(14,37,38,63,39,50,20,47)(15,41,45,65,42,58,26,56)(21,48,54,69,27,57,49,68)(36,62,55,67,40,64,46,66)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17977);