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×#23874 |
| 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 | |
1674 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)(8,13)(9,20)(10,12)(11,16)(14,21)(15,23)(17,32)(18,27)(19,29)(22,31)(24,26)(25,30)(28,33)(34,36)(35,37);(7,12)(10,23)(13,27)(14,20)(15,16)(18,24)(21,25)(26,29)(28,32)(30,31)(33,35)(36,37);(6,8,11,19)(7,13,16,29)(9,17,22,34)(10,24,23,18)(12,26,15,27)(14,28,30,37)(20,36,31,32)(21,35,25,33);(6,9)(7,14)(8,17)(10,25)(11,22)(12,20)(13,28)(15,31)(16,30)(18,35)(19,34)(21,23)(24,33)(26,36)(27,32)(29,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16862);