Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13334 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13334 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2011 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,19)(9,13)(10,26)(11,29)(12,17)(14,20)(15,23)(16,27)(18,25)(21,34)(22,31)(24,33)(28,36)(30,35)(32,37);(7,13)(9,23)(10,27)(11,25)(14,19)(15,18)(16,29)(17,26)(20,28)(21,30)(24,37)(31,32)(33,34)(35,36);(6,8)(7,14)(9,24)(10,28)(11,30)(12,22)(13,31)(15,34)(16,35)(17,32)(18,36)(19,26)(20,27)(21,25)(23,37)(29,33)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6322);