Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#9242 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C2×C5×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
478 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)(9,17,20,15)(10,16)(12,14)(13,19,21,18)(22,23)(24,27,31,34)(25,28)(26,29)(30,33,36,37)(32,35);(6,8,11,7,12,16,10,14)(9,15,18,13,20,17,19,21)(22,24,26,31)(23,27)(25,30,32,36)(28,33)(29,34)(35,37);(6,9,12,20)(7,13,14,21)(8,15,16,17)(10,19,11,18)(22,25,26,32)(23,28,29,35)(24,30,31,36)(27,33,34,37)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2230);