Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9213 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C5×#3106 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,10,13,11,14,9,19)(8,15,18,20,16,21,17,12)(23,27,29,32)(24,30,31,33);(6,8)(7,12,14,20)(9,17)(10,18)(11,16)(13,15,19,21)(22,23,25,32,26,29,34,27)(24,28,33,36,31,37,30,35);(8,16)(12,20)(15,21)(17,18)(22,24)(23,28)(25,33)(26,31)(27,35)(29,37)(30,34)(32,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2201);