Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×C5×#2511 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(4,6)(5,9)(12,15)(13,17)(19,20,23,26)(21,25,22,24);(3,4,7,6)(5,10,9,8)(11,12)(13,16)(14,15)(17,18)(19,21,23,22)(20,24,26,25);(3,5,7,9)(4,8,6,10)(19,21,23,22)(20,24,26,25);(3,6,7,4)(5,10,9,8)(11,13,14,17)(12,16,15,18)(19,22,23,21)(20,25,26,24)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27759);