Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24421 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,12,26,11,13,10,16)(8,17,25,14,20,28,18,27)(9,21,23,30,24,15,22,31)(19,32,33,36,34,37,35,29);(7,13)(8,18)(10,12)(14,28)(15,21)(17,27)(19,33)(20,25)(22,23)(29,37)(32,36)(34,35);(6,8)(7,14)(9,19)(10,25)(11,20)(12,18)(13,27)(15,29)(16,28)(17,26)(21,36)(22,33)(23,35)(24,34)(30,32)(31,37);(6,9)(7,15)(8,19)(10,23)(11,24)(12,22)(13,21)(14,29)(16,31)(17,32)(18,33)(20,34)(25,35)(26,30)(27,36)(28,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17409);