Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12980 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C22×C5×C16 ◁ G | |
470 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)(10,11,15,21,16,22,35,26,17,23,36,40,14,20,25,38)(12,24,19,39,27,13,29,31,28,30,33,41,32,18,37,34);(6,7,8,9)(10,12,25,29,14,32,36,19,17,28,35,37,16,27,15,33)(11,18,38,39,20,30,40,34,23,13,26,41,22,24,21,31);(10,13)(11,19)(12,26)(14,24)(15,34)(16,30)(17,18)(20,37)(21,32)(22,29)(23,33)(25,41)(27,40)(28,38)(31,36)(35,39)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5968);