Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19213 |
| 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 | |
594 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)(6,7,11,14)(8,15,13,10)(9,12,17,19)(16,21,20,18)(22,23)(24,30)(25,27)(26,34)(28,32)(29,33)(31,37)(35,36);(6,8,10,7,11,13,15,14)(9,16,18,12,17,20,21,19)(23,27)(28,36)(29,34)(30,37);(6,9)(7,12)(8,16)(10,18)(11,17)(13,20)(14,19)(15,21)(22,24,25,31)(23,28,27,36)(26,35,33,32)(29,37,34,30);(9,17)(12,19)(16,20)(18,21)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12201);