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) = C2×C4 |
| direct product | C5×#13271 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #476 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1987 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,16,12,17,26,36,13,18,25,15,11,28,9,24)(8,19,20,31,22,35,37,14,23,32,21,33,29,30,27,34);(6,8)(7,14,18,34)(9,20)(10,27)(11,22)(12,29)(13,23)(15,30,16,35)(17,31,28,33)(19,24,32,36)(21,26)(25,37);(6,9,11,25,13,26,12,10)(7,15,17,24,18,16,28,36)(8,20,22,37,23,21,29,27)(14,30,31,32,34,35,33,19)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6259);