Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#2889 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×C22⋊C4 |
| commutator | C22 | C23×C4×C5 |
| Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C2×C5×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ G | |
704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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)(6,7,10,15,11,14,23,28)(8,16,18,29,19,12,22,25)(9,13,21,27,20,26,33,36)(17,30,32,37,31,24,34,35);(6,8,11,19)(7,12,14,16)(9,17,20,31)(10,22,23,18)(13,24,26,30)(15,29,28,25)(21,34,33,32)(27,37,36,35);(6,9)(7,13)(8,17)(10,21)(11,20)(12,24)(14,26)(15,27)(16,30)(18,32)(19,31)(22,34)(23,33)(25,35)(28,36)(29,37);(9,20)(13,26)(17,31)(21,33)(24,30)(27,36)(32,34)(35,37)205 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18566);