Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#32872 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2826 |
| 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 | |
482 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)(6,7,12,17,13,19,11,18)(8,20,24,38,25,48,23,14)(9,26,30,15,31,41,29,42)(10,16,33,46,34,47,35,45)(21,36,27,39,53,62,52,49)(22,50,55,43,57,64,56,37)(28,44,32,40,58,66,59,67)(51,61,54,65,60,68,69,63);(6,8,11,23,13,25,12,24)(7,14,17,20,19,38,18,48)(9,27,29,21,31,52,30,53)(10,22,33,55,34,57,35,56)(15,39,41,62,42,49,26,36)(16,43,45,50,47,37,46,64)(28,54,32,60,58,69,59,51)(40,63,44,68,67,65,66,61);(6,9,13,31)(7,15,19,42)(8,21,25,53)(10,32,34,59)(11,30,12,29)(14,36,38,62)(16,44,47,66)(17,26,18,41)(20,49,48,39)(22,54,57,69)(23,27,24,52)(28,35,58,33)(37,63,43,65)(40,46,67,45)(50,68,64,61)(51,56,60,55);(6,10)(7,16)(8,22)(9,28)(11,35)(12,33)(13,34)(14,37)(15,40)(17,46)(18,45)(19,47)(20,50)(21,51)(23,56)(24,55)(25,57)(26,44)(27,54)(29,59)(30,32)(31,58)(36,61)(38,43)(39,65)(41,66)(42,67)(48,64)(49,63)(52,69)(53,60)(62,68)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25860);