Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11343 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C22×C5 | #385 |
| prime radical | C24×C5 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
666 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)(10,11,15,19,16,20,14,29)(12,21,25,32,24,37,23,17)(13,18,27,35,28,34,30,33)(22,36,38,41,26,40,39,31);(10,12,14,23,16,24,15,25)(11,17,19,21,20,32,29,37)(13,26,27,39,28,22,30,38)(18,31,33,40,34,41,35,36);(6,7,8,9)(10,13)(11,18)(12,22)(14,30)(15,27)(16,28)(17,31)(19,35)(20,34)(21,36)(23,39)(24,26)(25,38)(29,33)(32,41)(37,40)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4331);