Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24985 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24985 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | #2420 |
| prime radical | C5×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
646 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)(8,20)(9,24)(10,14)(11,18)(12,17)(13,19)(15,27)(16,23)(21,36)(22,31)(25,33)(26,32)(28,34)(29,35)(30,37);(6,8,12,22,13,23,11,10)(7,14,18,16,19,31,17,20)(9,25,27,21,29,26,28,37)(15,32,24,36,34,33,35,30);(6,9)(7,15)(8,21)(10,26)(11,27)(12,28)(13,29)(14,30)(16,33)(17,24)(18,35)(19,34)(20,32)(22,25)(23,37)(31,36);(6,10,11,23,13,22,12,8)(7,16,17,14,19,20,18,31)(9,26,27,37,29,25,28,21)(15,33,24,30,34,32,35,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17973);