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×#33382 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#578 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#41 ◁ C2×C5×#131 ◁ G | |
1124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7,8,9)(11,16)(13,23)(14,25)(17,29)(18,33)(19,30)(21,36)(22,35)(24,27)(26,28)(31,40)(32,41)(34,39)(37,38);(10,11,15,19)(12,17,20,31)(13,24,26,33)(14,27,25,18)(16,28,30,23)(21,37,34,41)(22,38,35,32)(29,39,40,36);(10,12)(11,17)(13,21)(14,22)(15,20)(16,29)(18,32)(19,31)(23,36)(24,37)(25,35)(26,34)(27,38)(28,39)(30,40)(33,41);(12,20)(17,31)(21,34)(22,35)(29,40)(32,38)(36,39)(37,41)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26370);