Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#13273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #476 |
| 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 | |
334 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)(7,9)(10,11,15,19,14,31,30,41,17,22,29,39,16,21,13,20)(12,23,27,18,33,36,32,35,28,34,24,38,26,40,25,37);(6,7,8,9)(10,12,14,26,17,28,16,33)(11,18,21,35,22,38,31,37)(13,24,15,32,30,27,29,25)(19,34,20,40,39,23,41,36);(10,13,16,29,17,30,14,15)(11,19,31,41,22,39,21,20)(12,24,33,25,28,27,26,32)(18,34,37,36,38,23,35,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6261);