Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C5×#2639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C22×C5×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
3326 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)(7,8,11,12)(9,10,13,14)(15,16)(17,20)(18,21)(19,22);(3,4,5,6)(8,12)(9,13)(15,17)(16,19)(18,22)(20,21);(3,4)(5,6)(7,9)(8,10)(11,13)(12,14)(15,17)(16,20)(18,22)(19,21);(7,10)(8,13)(9,12)(11,14)(15,16)(17,19)(18,21)(20,22)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27551);