Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18132 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18156 |
| 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 | |
982 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)(7,13)(9,22)(10,18)(14,30)(15,32)(16,29)(17,24)(19,36)(25,35)(27,28)(31,33)(34,37);(6,7)(8,14)(9,15)(10,24)(11,17)(12,16)(13,18)(19,33)(20,31)(21,28)(22,27)(23,32)(25,37)(26,34)(29,35)(30,36);(6,8,11,20)(7,14,17,31)(9,18,22,10)(12,21,26,23)(13,27,24,15)(16,28,34,32)(19,35,36,25)(29,30,37,33);(6,9,12,19)(7,15,16,33)(8,18,21,35)(10,23,25,20)(11,22,26,36)(13,28,29,14)(17,27,34,30)(24,32,37,31)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11144);