Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3544 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
870 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,14)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20)(22,23,26,29)(24,30,31,37)(25,33,34,28)(27,35,36,32);(6,8,10,16)(7,11,12,20)(9,18,19,15)(13,21,17,14)(25,34)(28,33)(30,37)(32,35);(6,7)(8,11)(9,13)(10,12)(14,15)(16,20)(17,19)(18,21)(22,24)(23,27)(25,35)(26,31)(28,37)(29,36)(30,33)(32,34);(7,12)(11,20)(13,17)(14,21)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11709);