Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20995 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#386 |
| prime radical | C2×C5×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
1664 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)(8,18)(9,22,25,31)(10,15)(11,16)(12,17)(13,20)(14,23,30,24)(19,33,36,27)(21,28)(26,29)(32,35,37,34);(6,8,10,20,12,21,11,26)(7,13,16,18,17,29,15,28)(9,19,23,34,25,36,24,35)(14,27,31,32,30,33,22,37);(6,9)(7,14)(8,19)(10,23)(11,24)(12,25)(13,27)(15,22)(16,31)(17,30)(18,32)(20,34)(21,36)(26,35)(28,37)(29,33);(6,10,12,11)(7,15,17,16)(8,20,21,26)(9,23,25,24)(13,28,29,18)(14,22,30,31)(19,34,36,35)(27,37,33,32)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13983);