Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#3022 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C2×#120 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C5×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(6,7,12,17)(8,18,21,30)(9,14,23,29)(10,16,11,15)(13,24,27,20)(19,33,36,32)(22,31,25,28)(26,34,37,35);(6,8)(7,13)(9,19)(10,20)(11,24)(12,21)(14,26)(15,18)(16,30)(17,27)(22,34)(23,36)(25,35)(28,32)(29,37)(31,33);(6,9)(7,14)(8,19)(10,22)(11,25)(12,23)(13,26)(15,28)(16,31)(17,29)(18,32)(20,34)(21,36)(24,35)(27,37)(30,33);(6,10,12,11)(7,15,17,16)(8,20,21,24)(9,22,23,25)(13,18,27,30)(14,28,29,31)(19,34,36,35)(26,32,37,33)145 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18699);