Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
514 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)(6,7,13,19)(8,20,25,53)(9,26,30,50)(10,31,36,48)(11,18,12,17)(14,38,42,22)(15,37,45,29)(16,34,49,35)(21,54,57,69)(23,52,24,46)(27,61,56,51)(28,47,33,44)(32,41,59,40)(39,65,68,62)(43,58,67,55)(60,66,64,63);(6,8)(7,14)(9,27)(10,32)(11,23)(12,24)(13,25)(15,43)(16,46)(17,40)(18,41)(19,42)(20,31)(21,29)(22,35)(26,39)(28,58)(30,56)(33,55)(34,38)(36,59)(37,57)(44,51)(45,67)(47,61)(48,53)(49,52)(50,68)(54,60)(62,63)(64,69)(65,66);(6,9)(7,15)(8,21)(10,33)(11,37)(12,29)(13,30)(14,39)(16,47)(17,50)(18,26)(19,45)(20,51)(22,58)(23,27)(24,56)(25,57)(28,36)(31,64)(32,65)(34,63)(35,66)(38,55)(40,43)(41,67)(42,68)(44,49)(46,69)(48,60)(52,54)(53,61)(59,62)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6312);