Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2223 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C2×C5×#472 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
658 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);(4,10)(5,16)(6,19)(7,8)(11,20)(12,27)(13,24)(14,23)(15,25)(17,32)(18,21)(22,28)(29,34)(30,33);(3,4,11,23,8,14,6,10,9,15,19,24,7,13,20,25)(5,12,26,34,16,28,17,22,18,29,31,27,21,30,32,33);(3,5)(4,12)(6,17)(7,21)(8,16)(9,18)(10,22)(11,26)(13,30)(14,28)(15,29)(19,31)(20,32)(23,34)(24,27)(25,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26604);