Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×C4×C5×#101 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C22⋊C4 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C2×C22⋊C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C22×C42×C5 ◁ G | |
1538 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);(7,8,10,14,11,13,19,21)(9,15,16,22,17,12,18,20);(7,9,11,17)(8,12,13,15)(10,18,19,16)(14,22,21,20)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13383);