Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23873 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2403 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1686 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)(9,20)(10,23)(11,15)(12,18)(13,21)(14,25)(17,28)(19,30)(22,31)(24,27)(26,29)(32,33)(34,36)(35,37);(6,8,11,19)(7,12,15,27)(9,17,22,34)(10,24,25,18)(13,28,29,36)(14,30,23,16)(20,32,31,37)(21,35,26,33);(10,25)(12,27)(14,23)(18,24)(20,31)(21,26)(28,36)(33,35);(6,9)(7,13)(8,17)(10,26)(11,22)(12,28)(14,31)(15,29)(16,32)(18,35)(19,34)(20,23)(21,25)(24,33)(27,36)(30,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16861);