Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21003 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,15,21,16,20,14,19)(12,17,23,31,24,33,25,32)(13,26,27,18,28,34,29,35)(22,36,38,30,39,40,37,41);(10,12,15,25,16,24,14,23)(11,17,19,31,20,33,21,32)(13,22,29,38,28,39,27,37)(18,30,34,36,35,41,26,40);(7,9)(10,13)(11,18)(12,22)(14,27)(15,29)(16,28)(17,30)(19,34)(20,35)(21,26)(23,37)(24,39)(25,38)(31,36)(32,40)(33,41);(10,14,16,15)(11,19,20,21)(12,23,24,25)(13,27,28,29)(17,31,33,32)(18,34,35,26)(22,37,39,38)(30,36,41,40)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13991);