Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1913 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#10757 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | C22⋊#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1168 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)(6,7,10,16,12,15,11,24)(8,13,19,27,18,26,23,28)(9,14,21,29,20,30,22,35)(17,31,33,37,32,25,34,36);(6,8,11,19,12,18,10,23)(7,13,16,28,15,26,24,27)(9,17,22,33,20,32,21,34)(14,25,29,37,30,31,35,36);(6,9)(7,14)(8,17)(10,21)(11,22)(12,20)(13,25)(15,30)(16,29)(18,32)(19,33)(23,34)(24,35)(26,31)(27,36)(28,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3745);