Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2085 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ C22×C42×C5 ◁ G | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4,5,6)(7,8)(9,14)(10,12)(11,13)(16,19)(20,22);(7,9,11,10)(8,12,13,14)(16,20)(18,21)(19,22);(4,6)(7,10,11,9)(8,12,13,14)(15,16)(17,19)(18,22)(20,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16642);