Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33369 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×Q8 | C24×C5 |
| socle | C2×C5 | C22×#99 |
| prime radical | C5×#3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
1220 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)(6,7,11,16,12,17,9,14)(8,18,20,35,19,33,26,37)(10,23,22,34,25,15,27,32)(13,28,31,24,29,36,30,21);(6,8)(7,13)(9,20)(10,24)(11,26)(12,19)(14,30)(15,33)(16,31)(17,29)(18,23)(21,25)(22,28)(27,36)(32,37)(34,35);(8,19)(13,29)(18,33)(20,26)(21,24)(28,36)(30,31)(35,37);(6,9,12,11)(7,14,17,16)(8,20,19,26)(10,22,25,27)(13,30,29,31)(15,32,23,34)(18,35,33,37)(21,36,24,28)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26357);