Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#20446 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C5×M16 ◁ G | |
1010 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(10,11)(12,20,23,33)(13,17)(14,19)(15,18)(16,22,29,26)(21,34,35,40)(24,31)(25,30)(27,32)(28,36,39,38)(37,41);(6,7,8,9)(10,12)(11,16)(13,21)(14,26)(15,23)(17,28)(18,29)(19,33)(20,32)(22,27)(24,35)(25,38)(30,40)(31,39)(34,41)(36,37);(10,13,14,25,15,24,27,37)(11,17,19,30,18,31,32,41)(12,21,26,38,23,35,22,36)(16,28,33,40,29,39,20,34);(11,18)(13,24)(16,29)(19,32)(20,33)(21,35)(25,37)(36,38)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13434);