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×#18659 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3514 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
938 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)(6,7,10,12)(8,14,15,21)(9,17,19,13)(11,18,20,16)(22,23,27,31)(24,32,35,36)(25,28,26,30)(29,34,37,33);(6,8)(7,11)(9,18)(10,15)(12,20)(13,21)(14,17)(16,19)(23,28)(25,26)(29,36)(30,31)(32,37)(33,34);(22,24)(23,29)(25,33)(26,34)(27,35)(28,36)(30,32)(31,37);(7,12)(8,15)(13,17)(16,18)(22,25,27,26)(23,30,31,28)(24,33,35,34)(29,32,37,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11647);