Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17968 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3171 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C5×#431 ◁ G | |
622 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,14)(13,17)(15,19)(16,18)(20,21)(22,23,26,30,27,24,31,29)(25,28,34,36,33,32,37,35);(7,11)(9,17)(12,18)(13,14)(16,21)(19,20)(22,24)(23,27)(25,32)(26,29)(28,33)(30,31)(34,35)(36,37);(6,8)(7,12)(9,16)(10,15)(11,18)(13,20)(14,19)(17,21)(22,25,26,34,27,33,31,37)(23,28,30,36,24,32,29,35);(8,15)(12,19)(16,21)(18,20)(23,24)(25,33)(29,30)(34,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10956);