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×#3487 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C22 | C26×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
11510 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)(8,9)(11,19)(13,15)(17,22);(7,8,12,9)(10,15,18,13)(11,14,19,16)(17,21,22,20);(3,4)(5,6)(7,9,12,8)(10,15,18,13)(11,16,19,14)(17,21,22,20);(7,10,12,18)(8,13,9,15)(11,17,19,22)(14,20,16,21);(7,11)(8,14)(9,16)(10,17)(12,19)(13,20)(15,21)(18,22)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 55777);