Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#32871 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
492 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)(6,7,12,14,11,17,10,16)(8,18,19,26,22,28,21,13)(9,15,23,29,25,32,24,31)(20,33,34,30,35,37,36,27);(6,8,10,21,11,22,12,19)(7,13,14,18,17,26,16,28)(9,20,23,34,25,35,24,36)(15,30,31,33,32,27,29,37);(7,14)(8,19)(9,23)(10,12)(13,26)(16,17)(20,35)(21,22)(24,25)(27,33)(29,31)(30,37);(6,9)(7,15)(8,20)(10,24)(11,25)(12,23)(13,27)(14,29)(16,31)(17,32)(18,33)(19,34)(21,36)(22,35)(26,30)(28,37)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25859);