Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2238 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#2238 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#469 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(4,8)(5,11)(6,14)(9,16)(10,13)(12,18)(15,17);(5,12)(8,16)(11,18)(13,17);(3,4,6,15,7,9,14,10)(5,13,11,16,12,17,18,8)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26642);