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×#3489 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
11438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 6 generators on 27 points:
(1,2)(23,24,25,26,27);(4,6)(7,8,12,13)(9,15,17,10)(11,19,20,14)(16,18,22,21);(8,13)(10,15)(14,19)(18,21);(3,4)(5,6)(7,9)(8,10)(11,16)(12,17)(13,15)(14,18)(19,21)(20,22);(7,10)(8,9)(11,18)(12,15)(13,17)(14,16)(19,22)(20,21);(7,11,12,20)(8,14,13,19)(9,16,17,22)(10,18,15,21)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55789);