Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11360 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #387 |
| 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 | |
576 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)(8,17)(9,22,26,32)(10,27)(11,15)(12,16)(13,19)(14,25,33,24)(18,34,36,28)(20,30)(21,29)(23,37,35,31);(6,8)(7,13,16,30)(9,23)(10,19)(11,20)(12,21)(14,31,33,37)(15,17,27,29)(18,24)(22,34,32,28)(25,36)(26,35);(6,9)(7,14,16,33)(8,18)(10,24)(11,25)(12,26)(13,28,30,34)(15,22,27,32)(17,31,29,37)(19,35)(20,23)(21,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4348);