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) = C24 |
| direct product | C5×#13028 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #459 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
476 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)(8,9)(10,11,16,23,36,37,35,21,18,25,15,22,17,24,14,34)(12,26,29,19,33,20,27,40,31,41,28,39,30,38,13,32);(10,12,17,33,18,31,36,30)(11,19,37,32,25,39,24,40)(13,14,29,15,27,35,28,16)(20,23,26,21,38,22,41,34);(7,9)(10,13)(11,20)(12,14)(15,33)(16,30)(17,29)(18,27)(19,23)(21,32)(22,39)(24,41)(25,38)(26,37)(28,36)(31,35)(34,40)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6016);