Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24423 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3586 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1284 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)(6,7,13,35,12,17,11,18)(8,19,33,38,23,14,22,44)(9,24,34,15,27,45,28,41)(10,29,31,42,32,16,30,43)(20,48,54,46,50,39,25,36)(21,37,52,63,53,47,51,62)(26,55,59,60,57,65,58,40)(49,67,56,61,68,64,69,66);(6,8,12,23)(7,14,17,19)(9,25,27,54)(10,21,32,53)(11,33,13,22)(15,39,41,48)(16,37,29,47)(18,44,35,38)(20,34,50,28)(24,36,45,46)(26,56,57,69)(30,52,31,51)(40,64,60,67)(42,63,43,62)(49,58,68,59)(55,61,65,66);(6,9)(7,15)(8,20)(10,26)(11,34)(12,27)(13,28)(14,36)(16,40)(17,41)(18,45)(19,46)(21,49)(22,54)(23,50)(24,35)(25,33)(29,60)(30,59)(31,58)(32,57)(37,61)(38,48)(39,44)(42,55)(43,65)(47,66)(51,56)(52,69)(53,68)(62,64)(63,67);(6,10)(7,16)(8,21)(9,26)(11,31)(12,32)(13,30)(14,37)(15,40)(17,29)(18,43)(19,47)(20,49)(22,52)(23,53)(24,55)(25,56)(27,57)(28,59)(33,51)(34,58)(35,42)(36,61)(38,63)(39,64)(41,60)(44,62)(45,65)(46,66)(48,67)(50,68)(54,69)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17411);