Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C5 · Schur multiplier (H2) = C42 |
| direct product | C5×#12541 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C42 |
| commutator | C4 | C43×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#97 |
| prime radical | C23×C5 | C2×M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
402 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)(10,11,13,17,14,19,25,33,15,18,26,32,27,34,40,41)(12,20,22,36,23,29,38,31,21,35,24,39,37,16,28,30);(10,12)(11,16,18,29)(13,24)(14,23)(15,21)(17,31,32,30)(19,20,34,35)(22,26)(25,28)(27,37)(33,39,41,36)(38,40);(6,7,8,9)(12,21)(16,29)(20,35)(22,24)(23,37)(28,38)(30,31)(36,39)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5529);