Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11285 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #124 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #394 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
606 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)(12,16,21,28)(13,17,24,31)(14,19)(15,18)(20,27)(22,29,35,40)(23,30)(25,32,37,38)(26,33)(34,39)(36,41);(10,12)(11,16,18,28)(13,23,24,36)(14,22)(15,21)(17,30)(19,29,33,40)(20,25,34,37)(26,35)(27,38)(31,41)(32,39);(10,13)(11,17,18,31)(12,20,21,34)(14,25)(15,24)(16,27)(19,32,33,38)(22,23,35,36)(26,37)(28,39)(29,30)(40,41)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4273);