Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11234 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11349 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #391 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1445 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
580 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)(8,12,17,27)(9,19,21,34)(10,15)(11,14)(13,20,29,36)(16,31)(18,26,22,30)(23,24)(25,35)(28,33)(32,37);(6,8,10,22,11,17,23,18)(7,12,24,26,14,27,15,30)(9,16,20,35,21,32,36,33)(13,28,34,37,29,25,19,31);(6,9,10,20,11,21,23,36)(7,13,24,34,14,29,15,19)(8,16,22,35,17,32,18,33)(12,25,26,31,27,28,30,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4337);