Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12933 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12933 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1853 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(7,13,16,27)(8,17)(9,20)(10,11)(14,28,30,35)(15,32,25,21)(18,34)(19,24)(22,23)(26,31,37,29);(6,7,11,25,12,16,10,15)(8,14,17,29,19,30,24,31)(9,21,20,13,23,32,22,27)(18,35,33,26,34,28,36,37);(6,8)(7,14)(9,18)(10,24)(11,17)(12,19)(13,26)(15,31)(16,30)(20,33)(21,35)(22,36)(23,34)(25,29)(27,37)(28,32)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5921);