Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×C5×#2363 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×#121 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
4046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(19,20,21,22,23);(4,6)(7,8,10,9)(11,12,15,14)(13,16,18,17);(7,9,10,8)(12,14)(16,17);(3,4)(5,6)(11,13,15,18)(12,16,14,17);(11,14,15,12)(13,17,18,16)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27123);