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×#18464 |
| 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 | |
970 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)(9,21)(13,26)(14,24)(15,22)(17,20)(19,34)(23,33)(25,30)(27,37)(28,36)(29,35)(31,32);(6,7,10,15)(8,13,17,27)(9,22,21,12)(11,14,18,28)(16,30,20,29)(19,35,32,25)(23,36,33,24)(26,31,37,34);(8,17)(9,23)(12,24)(13,27)(16,20)(19,31)(21,33)(22,36)(25,26)(29,30)(32,34)(35,37);(6,8,18,16)(7,13,28,30)(9,19,33,31)(10,17,11,20)(12,25,36,26)(14,29,15,27)(21,32,23,34)(22,35,24,37)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11452);