Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C2×C5×#1377 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C22⋊C4 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C23×C5 | C22⋊C8 |
| prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ C23×C5×C8 ◁ G | |
630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(23,24,25,26,27);(7,8,9,13,10,14,16,20,11,12,15,18,17,19,21,22);(3,4,5,6)(8,12)(9,15)(14,19)(16,21)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5538);