Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#19208 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C23×C5 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
606 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)(7,11)(9,12)(13,17)(14,18)(16,19)(20,21)(22,23,26,29)(24,30,32,37)(25,33,35,28)(27,34,36,31);(6,7,12,18,10,14,9,11)(8,13,19,21,15,20,16,17);(6,8)(7,13)(9,16)(10,15)(11,17)(12,19)(14,20)(18,21)(22,24,26,32)(23,27,29,36)(25,34,35,31)(28,37,33,30);(8,15)(13,20)(16,19)(17,21)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12196);