Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2082 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #397 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4,8,13,9,14,7,12)(5,15,18,10,19,26,17,27)(6,20,22,28,23,29,24,11)(16,31,32,34,21,25,33,30);(3,5,9,19)(4,10,14,27)(6,21,23,16)(7,18,8,17)(11,25,28,31)(12,26,13,15)(20,34,29,30)(22,32,24,33);(3,6,7,22,9,23,8,24)(4,11,12,20,14,28,13,29)(5,16,17,32,19,21,18,33)(10,25,15,30,27,31,26,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16639);