Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33381 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#576 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1176 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)(12,17)(13,16)(14,27)(15,20)(18,24)(19,25)(21,31)(22,28)(23,37)(26,30)(29,38)(32,36)(33,35)(34,39)(40,41);(11,16)(13,24)(14,25)(17,28)(18,19)(20,30)(22,36)(23,35)(26,38)(27,29)(31,39)(32,33)(34,41)(37,40);(10,12)(11,17)(13,22)(14,23)(15,21)(16,28)(18,32)(19,33)(20,31)(24,36)(25,35)(26,34)(27,37)(29,40)(30,39)(38,41);(12,21)(17,31)(22,34)(23,35)(28,39)(32,40)(33,37)(36,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26369);