Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11855 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
324 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)(8,14,16,11)(9,12)(10,13)(15,20,17,19)(18,21)(22,23)(24,34)(25,31)(26,41)(27,44)(28,33)(29,36)(30,39)(32,43)(35,47)(37,53)(38,49)(40,50)(42,51)(45,46)(48,52);(6,8)(7,11)(9,17)(10,16)(12,20)(13,14)(15,18)(19,21)(22,24)(23,29)(25,36)(26,42)(27,45)(28,38)(30,49)(31,46)(32,51)(33,48)(34,41)(35,39)(37,43)(40,52)(44,47)(50,53);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(22,25,27,39,28,40,43,26)(23,30,32,31,33,41,44,50)(24,35,37,36,38,42,45,52)(29,46,47,49,48,53,51,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4843);