Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2032 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23474 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #393 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| socle | C2×C5 | C2×#393 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1164 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)(9,19,22,13)(10,14)(11,15)(16,29,20,24)(17,25)(18,26)(21,32,34,27)(23,28)(30,37,33,35)(31,36);(6,8)(7,12)(9,20,22,16)(10,17)(11,18)(13,24,19,29)(14,25)(15,26)(21,33,34,30)(23,31)(27,35,32,37)(28,36);(6,9)(7,13)(8,16)(10,21)(11,22)(12,24)(14,27)(15,19)(17,30)(18,20)(23,34)(25,35)(26,29)(28,32)(31,33)(36,37);(6,10,11,23)(7,14,15,28)(8,17,18,31)(9,21,22,34)(12,25,26,36)(13,27,19,32)(16,30,20,33)(24,35,29,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16462);