Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1146 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)(9,13)(10,23)(11,15)(12,20)(14,26)(17,24)(18,34)(19,30)(21,31)(22,29)(25,28)(27,36)(32,35)(33,37);(6,7)(8,17,19,12)(9,13)(10,14)(11,15)(16,20,30,24)(18,33,35,27)(21,28)(22,29)(23,26)(25,31)(32,37,34,36);(6,8)(7,12)(9,18)(10,24)(11,19)(13,27)(14,30)(15,17)(16,23)(20,26)(21,37)(22,35)(25,36)(28,32)(29,33)(31,34);(6,9,11,22)(7,13,15,29)(8,18,19,35)(10,25,26,21)(12,27,17,33)(14,31,23,28)(16,32,30,34)(20,37,24,36)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16987);