Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#24987 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2420 |
| 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 | |
638 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)(10,22)(14,29)(15,25)(18,19)(21,33)(23,27)(26,34)(30,31)(36,37);(6,7,10,12,11,15,22,25)(8,13,16,27,19,28,18,23)(9,14,20,24,21,29,33,35)(17,26,30,36,32,37,31,34);(6,8)(7,13)(9,17)(10,18)(11,19)(12,23)(14,26)(15,28)(16,22)(20,31)(21,32)(24,34)(25,27)(29,37)(30,33)(35,36);(6,9,10,20,11,21,22,33)(7,14,12,24,15,29,25,35)(8,17,18,31,19,32,16,30)(13,26,23,34,28,37,27,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17975);