Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11688 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #327 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| socle | C2×C5 | C2×#327 |
| prime radical | C5×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
432 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)(6,7,9,14,12,16,11,27)(8,17,20,31,21,35,18,34)(10,23,25,15,26,32,22,33)(13,24,28,37,30,36,29,19);(6,8)(7,13)(9,18)(10,24)(11,20)(12,21)(14,28)(15,31)(16,30)(17,23)(19,25)(22,37)(26,36)(27,29)(32,35)(33,34);(6,9,12,11)(7,14,16,27)(8,18,21,20)(10,22,26,25)(13,28,30,29)(15,23,33,32)(17,34,35,31)(19,24,37,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4676);