Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1953 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #121 |
| commutator | C2×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #389 |
| prime radical | C23×C5 | C2×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
184 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,15,12,16,10,26)(8,17,19,13,21,28,20,29)(9,22,24,30,25,36,23,14)(18,27,32,35,34,37,33,31);(6,8,11,19,12,21,10,20)(7,13,15,28,16,29,26,17)(9,18,24,32,25,34,23,33)(14,27,22,35,30,37,36,31);(6,9,10,23,12,25,11,24)(7,14,15,22,16,30,26,36)(8,18,20,33,21,34,19,32)(13,27,28,35,29,37,17,31)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5543);