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) = C26 |
| direct product | C2×C5×#2143 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×D8 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C22×C5 | C2×#102 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,7,12,9,13,8,23)(5,14,17,31,15,30,24,28)(6,18,20,29,22,33,21,11)(10,19,26,32,25,34,27,16);(3,5)(4,10)(6,19)(7,17)(8,24)(9,15)(11,28)(12,26)(13,25)(14,18)(16,21)(20,32)(22,34)(23,27)(29,31)(30,33);(5,15)(10,25)(14,30)(16,32)(17,24)(19,34)(26,27)(28,31)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26372);