Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C4×C5×#510 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C22×D8 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
674 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)(37,38,39,40,41);(5,6)(7,15)(8,19)(9,24)(10,12)(11,16)(13,22)(14,21)(17,35)(18,32)(20,30)(23,31)(25,29)(26,34)(27,36)(28,33);(5,7,10,18)(6,11,12,26)(8,20,23,36)(9,25,21,17)(13,29,28,35)(14,30,24,27)(15,31,32,19)(16,33,34,22);(6,12)(11,26)(13,28)(14,24)(15,32)(19,31)(27,30)(29,35)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26441);