Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#9327 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #386 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
492 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)(6,7,10,13,11,14,9,19)(8,15,18,20,16,21,17,12)(22,23)(24,27,31,35)(25,28)(26,29)(30,33)(32,36,37,34);(6,8)(7,12,14,20)(9,17)(10,18)(11,16)(13,15,19,21)(22,24)(23,27,29,35)(25,32)(26,31)(28,36,30,34)(33,37);(8,16)(12,20)(15,21)(17,18)(23,28)(24,31)(25,33)(27,34)(29,30)(35,36)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2315);