Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2216 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C24×C5 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ C23×C5×C8 ◁ G | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2)(39,40,41,42,43);(3,4)(5,6)(7,8,12,16)(9,17,21,33)(10,22,24,37)(11,25,23,15)(13,26,28,19)(14,29,30,18)(20,36,35,32)(27,34,31,38);(7,9,18,34,11,20,24,26,12,21,29,38,23,35,10,19)(8,13,22,32,15,27,30,33,16,28,37,36,25,31,14,17);(4,6)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26586);