Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#34638 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7,9,10)(8,11,12,13)(14,15)(16,18)(17,19)(20,21)(22,23)(24,27)(25,29)(26,28)(30,33)(31,34)(32,35)(36,37);(7,10)(11,13)(14,16,17,20)(15,18,19,21)(22,24)(23,27)(25,31)(26,30)(28,33)(29,34)(32,36)(35,37);(6,8,9,12)(7,11,10,13)(15,19)(18,21)(24,30)(27,33)(31,36)(34,37);(23,28)(27,33)(29,35)(34,37);(22,25,26,32)(23,29,28,35)(24,31,30,36)(27,34,33,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27626);