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×#24000 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2404 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1170 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)(8,12)(9,19)(10,23)(11,15)(13,22)(14,25)(16,31)(17,33)(18,28)(20,36)(21,30)(24,29)(26,32)(27,35)(34,37);(6,8,11,18)(7,12,15,28)(9,20)(10,17,25,35)(13,26)(14,27,23,33)(16,21)(19,31)(22,34)(24,32)(29,37)(30,36);(6,9)(7,13)(8,16)(10,24)(11,21)(12,26)(14,30)(15,29)(17,34)(18,20)(19,23)(22,25)(27,36)(28,37)(31,33)(32,35);(10,25)(14,23)(17,35)(19,30)(22,24)(27,33)(31,36)(32,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16988);