Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3506 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
930 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)(6,7,11,14)(8,15,18,20)(9,13,10,19)(12,17,21,16)(22,23,27,31)(24,32,33,36)(25,30,26,29)(28,35,37,34);(6,8)(7,12)(9,16)(10,17)(11,18)(13,20)(14,21)(15,19)(22,24)(23,28)(25,35)(26,34)(27,33)(29,36)(30,32)(31,37);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,20,21,15)(23,29)(25,26)(28,36)(30,31)(32,37)(34,35);(6,9,11,10)(7,13,14,19)(8,17,18,16)(12,15,21,20)(24,33)(28,37)(32,36)(34,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11603);