Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
414 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)(7,9)(10,11)(12,22)(13,26,28,33)(14,30)(15,17)(16,21)(18,24,36,40)(19,23)(20,25)(27,39,29,38)(31,35,41,37)(32,34);(6,7,8,9)(10,12,23,32,16,25,14,15)(11,17,30,20,21,34,19,22)(13,24,29,41,28,40,27,31)(18,33,37,38,36,26,35,39);(10,13)(11,18)(12,24)(14,27)(15,31)(16,28)(17,33)(19,35)(20,38)(21,36)(22,39)(23,29)(25,40)(26,34)(30,37)(32,41)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5812);