Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#11132 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #397 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1439 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
864 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)(9,22,26,14)(10,27)(11,15)(12,16)(13,19)(18,33,36,30)(20,34)(21,29)(23,28,35,37)(24,31,25,32);(6,8,10,19,12,20,11,21)(7,13,15,17,16,29,27,34)(9,23,24,18,26,35,25,36)(14,30,31,37,22,33,32,28);(6,9)(7,14)(8,18)(10,25)(11,24)(12,26)(13,28)(15,32)(16,22)(17,33)(19,23)(20,36)(21,35)(27,31)(29,37)(30,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4120);