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×#2357 |
| 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 | |
3158 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)(7,8,11,10)(9,13,14,12)(15,16,20,19)(17,22,18,21);(8,10)(12,13)(15,17,20,18)(16,21,19,22);(7,9)(8,12)(10,13)(11,14)(15,18,20,17)(16,21,19,22);(7,10,11,8)(9,13,14,12)(15,19,20,16)(17,21,18,22)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27108);