Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19401 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3207 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
632 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)(8,15,18,21)(9,13,10,19)(12,16,20,17)(24,30)(25,32)(27,34)(29,36)(31,33)(35,37);(6,8,10,17,11,18,9,16)(7,12,13,15,14,20,19,21)(22,23)(24,27)(25,29)(26,28)(30,34)(31,35)(32,36)(33,37);(22,24,26,31)(23,27,28,35)(25,33,32,30)(29,37,36,34);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)(23,28)(27,35)(29,36)(34,37)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12389);