Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C5×#2353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
3374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,21)(18,20)(19,22);(8,11)(12,14)(15,16,18,20)(17,19,22,21);(7,9,10,13)(8,12,11,14)(15,17)(16,19)(18,22)(20,21);(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,20)(21,22)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27104);