Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13313 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(10,11,17,37,18,24,35,23)(12,25,30,38,13,31,33,40)(14,34,26,21,15,22,16,36)(19,32,41,28,20,29,39,27);(6,7)(8,9)(10,12,26,32,17,30,15,28,18,13,16,29,35,33,14,27)(11,19,34,40,23,39,36,31,24,20,22,38,37,41,21,25);(7,9)(10,13,18,12)(11,20,24,19)(14,29,15,32)(16,27,26,28)(17,30,35,33)(21,38,36,40)(22,25,34,31)(23,39,37,41)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6459);