Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13180 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(6,7,9,8)(10,11)(12,22)(13,19)(14,29)(15,32)(16,21)(17,23)(18,27)(20,31)(24,37)(25,39)(26,33)(28,38)(30,34)(35,40)(36,41);(6,8,9,7)(10,12)(11,17)(13,23)(14,30)(15,24)(16,26)(18,33)(19,37)(20,34)(21,36)(22,29)(25,31)(27,40)(28,41)(32,35)(38,39);(10,13,15,27,16,28,31,14)(11,18,20,19,21,29,32,38)(12,23,24,40,26,41,25,30)(17,33,34,37,36,22,35,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6168);