Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20892 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#385 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1320 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)(6,7,10,15,11,14,21,27)(8,12,18,24,17,23,31,35)(9,13,20,26,19,25,32,36)(16,28,30,37,29,22,33,34);(6,8,10,18,11,17,21,31)(7,12,15,24,14,23,27,35)(9,16,20,30,19,29,32,33)(13,22,26,34,25,28,36,37);(6,9)(7,13)(8,16)(10,20)(11,19)(12,22)(14,25)(15,26)(17,29)(18,30)(21,32)(23,28)(24,34)(27,36)(31,33)(35,37);(7,14)(12,23)(13,25)(15,27)(22,28)(24,35)(26,36)(34,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13880);