Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19057 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
908 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)(6,7,11,14)(8,15,17,20)(9,13,10,18)(12,19,21,16)(22,23,26,29)(24,30,31,37)(25,33,34,28)(27,35,36,32);(6,8,11,17)(7,12,14,21)(9,16,10,19)(13,20,18,15)(25,34)(28,33)(30,37)(32,35);(9,10)(13,18)(15,20)(16,19)(22,24)(23,27)(25,35)(26,31)(28,37)(29,36)(30,33)(32,34);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12045);