Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#18459 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
972 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)(7,12)(8,16)(9,20)(13,26)(14,24)(15,21)(17,19)(18,33)(22,35)(25,30)(27,37)(28,34)(29,36)(31,32);(6,7,10,15)(8,13,17,27)(9,21,20,12)(11,14,23,29)(16,30,19,28)(18,34,32,25)(22,36,35,24)(26,31,37,33);(8,17)(9,22)(12,24)(13,27)(16,19)(18,31)(20,35)(21,36)(25,26)(28,30)(32,33)(34,37);(6,8,11,19)(7,13,14,28)(9,18,22,33)(10,17,23,16)(12,25,24,37)(15,27,29,30)(20,32,35,31)(21,34,36,26)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11447);