Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24410 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2410 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1322 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)(8,16,19,12)(9,21,22,13)(10,15)(11,14)(17,24,28,20)(18,27)(23,34,35,29)(25,26)(30,33)(31,36)(32,37);(6,8,11,19)(7,12,14,16)(9,18,22,33)(10,24,26,20)(13,27,21,30)(15,17,25,28)(23,36,35,32)(29,31,34,37);(6,7)(8,17)(9,13)(10,25)(11,14)(12,24)(15,26)(16,20)(18,31)(19,28)(21,22)(23,34)(27,36)(29,35)(30,32)(33,37);(6,9)(7,13)(8,18)(10,23)(11,22)(12,27)(14,21)(15,29)(16,30)(17,31)(19,33)(20,32)(24,36)(25,34)(26,35)(28,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17398);