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×#18519 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
914 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)(6,7,10,13)(8,14,16,21)(9,17,19,12)(11,18,20,15)(22,23,28,33)(24,30,35,36)(25,29,34,37)(26,31,27,32);(6,8,10,16)(7,11,13,20)(9,18,19,15)(12,21,17,14)(22,24,28,35)(23,29,33,37)(25,26,34,27)(30,32,36,31);(22,25)(23,30)(24,26)(27,35)(28,34)(29,32)(31,37)(33,36);(22,26,28,27)(23,31,33,32)(24,34,35,25)(29,30,37,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11507);