Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#21009 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1708 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)(9,22,24,14)(10,15)(11,25)(12,16)(13,20)(18,31,34,37)(19,28)(21,32)(23,30,26,29)(27,33,36,35);(6,8,11,20,12,21,10,19)(7,13,15,17,16,28,25,32)(9,18,26,35,24,34,23,33)(14,27,29,31,22,36,30,37);(6,9)(7,14)(8,18)(10,23)(11,26)(12,24)(13,27)(15,29)(16,22)(17,31)(19,33)(20,35)(21,34)(25,30)(28,36)(32,37);(6,10,12,11)(7,15,16,25)(8,19,21,20)(9,23,24,26)(13,17,28,32)(14,29,22,30)(18,33,34,35)(27,31,36,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13997);