Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#11749 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #102 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C5×#466 | C22 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(7,11,12,13)(14,15,18,23,20,24,19,34)(16,25,27,41,28,42,35,39)(17,29,31,40,33,44,32,22)(21,30,36,43,38,45,37,26);(6,7)(8,11)(9,13)(10,12)(14,16)(15,21)(17,30)(18,27)(19,35)(20,28)(22,39)(23,36)(24,38)(25,29)(26,32)(31,43)(33,45)(34,37)(40,41)(42,44);(6,8,10,9)(7,11,12,13)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4737);