Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2404 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C5×#3026 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C24×C5 ◁ G | |
1210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,13)(10,14)(11,15)(16,23)(17,24)(18,25)(19,31)(20,33)(21,27)(22,28)(26,34)(29,37)(30,36)(32,35);(6,8,11,18)(7,12,15,25)(9,19)(10,17,22,30)(13,23)(14,24,28,36)(16,21)(20,32)(26,35)(27,31)(29,34)(33,37);(6,9)(7,13)(8,16)(10,20)(11,21)(12,23)(14,26)(15,27)(17,29)(18,19)(22,34)(24,35)(25,31)(28,33)(30,32)(36,37);(6,10)(7,14)(8,17)(9,20)(11,22)(12,24)(13,26)(15,28)(16,29)(18,30)(19,32)(21,34)(23,35)(25,36)(27,33)(31,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16985);