Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×C5×#2833 |
| 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 | |
1718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2)(35,36,37,38,39);(4,7)(8,10)(11,12,15,16)(13,17,18,14)(20,24)(25,27)(26,30)(33,34);(5,9)(8,10)(13,18)(14,17)(19,20,23,24)(21,27,29,25)(22,30,32,26)(28,33,31,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,18)(16,17)(19,21,23,29)(20,25,24,27)(22,31,32,28)(26,34,30,33);(3,5)(4,8)(6,9)(7,10)(11,14)(12,13)(15,17)(16,18)(19,22,23,32)(20,26,24,30)(21,28,29,31)(25,33,27,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29656);