Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18682 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18711 |
| 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 | |
818 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)(8,15)(9,13)(10,18)(11,14)(12,16)(17,20)(19,21)(22,23,28,33)(24,34,30,37)(25,36,29,35)(26,31,27,32);(6,7)(8,12)(9,13)(10,18)(11,14)(15,19)(16,20)(17,21)(22,24,28,30)(23,29,33,25)(26,35,27,36)(31,34,32,37);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,25,28,29)(23,30,33,24)(26,34,27,37)(31,35,32,36);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,26,28,27)(23,31,33,32)(24,35,30,36)(25,37,29,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11699);