Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#8603 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #117 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #380 |
| prime radical | C24×C5 | C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
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)(24,32)(25,36,38,29)(26,39)(27,31)(28,34)(30,40)(33,46,48,44)(35,45)(37,43,52,51)(41,47,50,53)(42,49);(6,8)(7,11)(9,15)(10,14)(12,18)(13,17)(16,20)(19,21)(22,24,26,34,27,35,40,49)(23,28,30,32,31,42,39,45)(25,33,37,47,38,48,52,53)(29,41,43,44,36,50,51,46);(8,14)(11,17)(15,20)(18,21)(22,25)(23,29)(24,33)(26,37)(27,38)(28,41)(30,43)(31,36)(32,44)(34,47)(35,48)(39,51)(40,52)(42,50)(45,46)(49,53)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1591);