Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11651 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11656 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| prime radical | C5×#466 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7,9,12,10,13,16,19)(8,11,15,18,14,17,20,21)(22,23)(24,33,36,49)(25,29)(26,40)(27,32)(28,35,44,53)(30,38)(31,39)(34,47,37,50)(41,43,51,45)(42,46)(48,52);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24)(23,28)(25,37)(26,41)(27,36)(29,45)(30,47)(31,43)(32,44)(33,46)(34,39)(35,52)(38,53)(40,49)(42,51)(48,50);(8,14)(11,17)(15,20)(18,21)(22,25,27,39)(23,29,32,31)(24,34,36,37)(26,38,42,52)(28,43,44,45)(30,46,48,40)(33,47,49,50)(35,51,53,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4644);