Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33347 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| socle | C2×C5 | C22×#99 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
1236 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)(8,16)(9,13)(10,22,23,14)(11,15)(12,19,24,31)(17,28)(18,29)(20,33,34,26)(21,27)(25,32,35,37)(30,36);(6,8)(7,12)(9,18)(10,19)(11,17)(13,25)(14,16)(15,24)(20,32)(21,30)(22,28)(23,31)(26,29)(27,35)(33,36)(34,37);(8,17)(12,24)(16,28)(18,30)(19,31)(25,35)(29,36)(32,37);(6,9,11,21)(7,13,15,27)(8,18,17,30)(10,20,23,34)(12,25,24,35)(14,26,22,33)(16,29,28,36)(19,32,31,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26335);