Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23892 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23892 |
| 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)(8,18)(9,24)(10,28)(11,16)(12,17)(13,21)(14,26)(15,25)(19,35)(20,30)(22,32)(23,31)(27,34)(29,36)(33,37);(6,8)(7,13)(9,20)(10,22)(11,21)(12,23)(14,33)(15,30)(16,31)(17,32)(18,28)(19,25)(24,29)(26,36)(27,37)(34,35);(6,9)(7,14)(8,19)(10,25)(11,26)(12,27)(13,29)(15,17)(16,34)(18,30)(20,21)(22,37)(23,36)(24,28)(31,33)(32,35);(6,9)(7,15)(8,20)(10,26)(11,25)(12,27)(13,30)(14,17)(16,34)(18,29)(19,21)(22,36)(23,37)(24,28)(31,35)(32,33)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16880);