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×#24420 |
| 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 | |
1326 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)(6,7,10,16,12,13,11,17)(8,18,22,27,21,29,19,14)(9,23,24,31,25,15,26,30)(20,32,33,28,34,37,35,36);(7,13)(8,19)(10,11)(14,18)(15,23)(20,33)(21,22)(24,26)(27,29)(28,32)(34,35)(36,37);(6,8)(7,14)(9,20)(10,19)(11,22)(12,21)(13,27)(15,28)(16,29)(17,18)(23,36)(24,35)(25,34)(26,33)(30,32)(31,37);(6,9)(7,15)(8,20)(10,26)(11,24)(12,25)(13,23)(14,28)(16,31)(17,30)(18,32)(19,33)(21,34)(22,35)(27,36)(29,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17408);