Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9159 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9178 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | #120 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,11,15,21)(9,14)(10,13)(12,16,20,17)(18,19)(22,23,26,30,27,31,25,29)(24,32,33,28,35,36,34,37);(6,8)(7,11,13,21)(9,17)(10,15)(12,19,20,14)(16,18)(22,24)(23,28)(25,33)(26,34)(27,35)(29,36)(30,32)(31,37);(6,8)(7,12)(9,17)(10,15)(11,19)(13,20)(14,21)(16,18)(22,25,27,26)(23,29,31,30)(24,33,35,34)(28,36,37,32)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2166);