Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21782 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21783 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3499 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17,20,12)(9,21,23,14)(10,16)(11,15)(13,19,29,24)(18,26)(22,35,37,30)(25,27)(28,32)(31,33)(34,36);(6,8)(7,12)(9,18)(10,24)(11,20)(13,27)(14,26)(15,17)(16,29)(19,25)(21,31)(22,36)(23,33)(28,35)(30,34)(32,37);(6,8)(7,13)(9,18)(10,24)(11,20)(12,16)(14,28)(15,29)(17,27)(19,25)(21,34)(22,36)(23,33)(26,30)(31,35)(32,37);(6,9,11,23)(7,14,15,21)(8,18,20,33)(10,22,25,37)(12,26,17,31)(13,28,29,34)(16,30,27,35)(19,32,24,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14771);