Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C5 · Schur multiplier (H2) = C214 |
| direct product | C2×C5×#3499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
11126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2)(23,24,25,26,27);(4,6)(7,8,11,14)(9,12,17,15)(10,18,19,13)(16,21,22,20);(7,8)(9,15)(10,13)(11,14)(12,17)(16,21)(18,19)(20,22);(3,4,5,6);(7,9)(8,12)(10,16)(11,17)(13,20)(14,15)(18,21)(19,22);(7,10,11,19)(8,13,14,18)(9,16,17,22)(12,20,15,21)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55790);