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) = C24×C4 |
| direct product | C2×C4×C5×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5×C8 | C22 |
| commutator | C2 | C22×C4×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C23×C5 | C22×C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5×C8 ◁ G | |
918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4,5,6);(7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(8,16)(10,18)(12,20)(14,22)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26310);