Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19014 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
824 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)(6,7,11,9)(8,14,16,21)(10,18,20,13)(12,19,15,17)(24,30)(25,32)(27,35)(28,33)(31,34)(36,37);(6,8,11,16)(7,12,9,15)(10,19,20,17)(13,21,18,14)(22,23,26,29)(24,27,31,36)(25,33,32,28)(30,37,34,35);(6,9,11,7)(8,15,16,12)(10,13,20,18)(14,19,21,17)(22,24)(23,27)(25,34)(26,31)(28,37)(29,36)(30,32)(33,35);(8,16)(12,15)(14,21)(17,19)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12017);