Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18521 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C5×#3480 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
936 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)(22,23,28,33)(24,34,36,30)(25,37,35,29)(26,31,27,32);(6,8)(7,12)(9,16)(10,19)(11,17)(13,20)(14,21)(15,18)(22,24)(23,29)(25,26)(27,35)(28,36)(30,32)(31,34)(33,37);(6,9,11,10)(7,13,14,18)(8,16,17,19)(12,20,21,15)(22,25,28,35)(23,30,33,34)(24,26,36,27)(29,32,37,31);(22,26,28,27)(23,31,33,32)(24,25,36,35)(29,34,37,30)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11509);