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) = C24 |
| direct product | C5×#11307 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #397 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1443 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
548 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)(8,12,17,26)(9,19)(10,23)(11,14)(13,21)(15,24)(16,25,30,37)(18,32,31,27)(20,33,35,28)(22,29)(34,36);(6,8)(7,12,14,26)(9,20,21,35)(10,18)(11,17)(13,28)(15,27,23,32)(16,22,30,36)(19,33)(24,31)(25,34)(29,37);(6,9)(7,13,14,19)(8,16,17,30)(10,22)(11,21)(12,25)(15,29,23,34)(18,20,31,35)(24,36)(26,37)(27,28)(32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4295);