Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9048 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #386 |
| prime radical | C23×C5 | C4×D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
424 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)(6,7,10,12,11,15,9,14)(8,16,18,13,19,20,17,21)(22,23,26,30,27,24,28,32)(25,31,34,37,33,29,35,36);(7,12)(9,10)(13,16)(14,15)(17,18)(20,21)(22,24,26,32,27,23,28,30)(25,31,35,36,33,29,34,37);(6,8,11,19)(7,13,15,21)(9,18,10,17)(12,16,14,20)(22,25)(23,29)(24,31)(26,35)(27,33)(28,34)(30,37)(32,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2036);