Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10988 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1274 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)(12,20)(13,24,26,17)(14,27)(15,19)(16,22)(18,30)(21,38,40,37)(23,35)(25,34,29,36)(28,32)(31,41,33,39);(10,12,15,23)(11,16,19,32)(13,21,26,40)(14,28,30,22)(17,33,24,31)(18,35,27,20)(25,41,29,39)(34,38,36,37);(7,9)(10,13)(11,17)(12,21)(14,29)(15,26)(16,31)(18,36)(19,24)(20,37)(22,41)(23,40)(25,30)(27,34)(28,39)(32,33)(35,38)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3976);