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) = C25×C4 |
| direct product | C5×#33358 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | 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 | |
1252 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)(8,16,17,29)(9,13)(10,22,24,14)(11,15)(12,19)(18,30,31,37)(20,33,35,27)(21,28)(23,25)(26,32)(34,36);(6,8)(7,12)(9,18)(10,23)(11,17)(13,26)(14,29)(15,25)(16,22)(19,24)(20,34)(21,31)(27,37)(28,36)(30,33)(32,35);(8,17)(12,25)(16,29)(18,31)(19,23)(26,36)(30,37)(32,34);(6,9,11,21)(7,13,15,28)(8,18,17,31)(10,20,24,35)(12,26,25,36)(14,27,22,33)(16,30,29,37)(19,32,23,34)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26346);