Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12743 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1786 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
926 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)(6,7,10,17,12,16,11,28)(8,18,20,34,21,14,27,33)(9,22,24,35,26,29,25,15)(13,23,31,36,30,37,32,19);(6,8)(7,13)(9,23)(10,20)(11,27)(12,21)(14,29)(15,33)(16,30)(17,31)(18,22)(19,25)(24,36)(26,37)(28,32)(34,35);(6,8)(7,14)(9,19)(10,27)(11,20)(12,21)(13,29)(15,32)(16,18)(17,34)(22,30)(23,25)(24,37)(26,36)(28,33)(31,35)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5731);