Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12745 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #418 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1786 |
| prime radical | C5×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
890 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)(8,18)(9,23,25,14)(10,27)(11,33)(12,16)(13,20)(15,31)(17,30)(19,44,47,65)(21,45)(22,51)(24,54,29,62)(26,39,56,53)(28,43)(32,59)(34,40)(35,46,52,49)(36,61)(37,50)(38,57,41,63)(42,64)(48,67,58,69)(55,66,60,68);(6,8)(7,13)(9,19)(10,28)(11,34)(12,21)(14,35)(15,40)(16,37)(17,43)(18,27)(20,31)(22,30)(23,52)(24,55)(25,47)(26,58)(29,60)(32,61)(33,36)(38,67)(39,68)(41,69)(42,51)(44,54)(45,59)(46,57)(48,56)(49,63)(50,64)(53,66)(62,65);(6,9)(7,14)(8,19)(10,29)(11,26)(12,25)(13,35)(15,41)(16,23)(17,39)(18,44)(20,49)(21,47)(22,48)(24,32)(27,54)(28,60)(30,56)(31,63)(33,53)(34,58)(36,66)(37,52)(38,42)(40,69)(43,68)(45,65)(46,50)(51,67)(55,61)(57,64)(59,62)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5733);