Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21483 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C4×#120 |
| prime radical | C5×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
1328 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)(6,7,11,16,12,10,15,24)(8,13,20,27,19,18,26,33)(9,14,23,29,22,21,28,34)(17,30,32,37,31,25,35,36);(6,8)(7,13)(9,17)(10,18)(11,20)(12,19)(14,25)(15,26)(16,27)(21,30)(22,31)(23,32)(24,33)(28,35)(29,36)(34,37);(6,9)(7,14)(8,17)(10,21)(11,23)(12,22)(13,25)(15,28)(16,29)(18,30)(19,31)(20,32)(24,34)(26,35)(27,36)(33,37);(6,10,11,24,12,7,15,16)(8,18,20,33,19,13,26,27)(9,21,23,34,22,14,28,29)(17,30,32,37,31,25,35,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14471);