Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5×C8 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×C5×C8×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C5×C8 | C22 |
| commutator | C2 | C22×C4×C5×C8 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C24×C5 | C22×C4 |
| prime radical | C25×C5 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5×C8 ◁ G | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,5,6,7,8,9,10);(11,12,14,16)(13,15,17,18);(12,15)(16,18)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13368);