Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61109 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#61111 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×D42 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
2600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,17)(9,21)(10,14)(11,28)(12,15)(13,20)(16,29)(18,35)(19,23)(22,24)(25,34)(26,33)(27,32)(30,37)(31,36);(6,8)(7,13)(9,15)(10,24)(11,21)(12,19)(14,31)(16,23)(17,28)(18,25)(20,29)(22,34)(26,36)(27,37)(30,33)(32,35);(6,9,12,13)(7,8,15,19)(10,22,27,30)(11,23,29,17)(14,18,32,36)(16,20,28,21)(24,34,37,33)(25,35,26,31);(6,10)(7,14)(8,18)(9,22)(11,26)(12,27)(13,30)(15,32)(16,34)(17,35)(19,36)(20,37)(21,24)(23,31)(25,29)(28,33);(7,15)(8,19)(14,32)(16,28)(18,36)(20,21)(24,37)(33,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 54099);