Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18628 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3519 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,17)(10,12)(11,16)(13,19)(15,21)(18,20)(22,23,27,30)(24,32,33,36)(25,31,26,29)(28,35,37,34);(6,8)(7,11)(9,18)(10,15)(12,20)(13,21)(14,17)(16,19)(22,24)(23,28)(25,35)(26,34)(27,33)(29,36)(30,37)(31,32);(23,29)(25,26)(28,36)(30,31)(32,37)(34,35);(7,12)(8,15)(13,17)(16,18)(23,30)(24,33)(29,31)(34,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11616);