Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#1791 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #120 |
| commutator | C23 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #379 |
| prime radical | C5×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
548 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 132 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)(6,7,10,15,11,14,23,28)(8,16,19,29,18,12,22,25)(9,13,21,26,20,27,33,37)(17,24,32,35,31,30,34,36);(6,8)(7,12)(9,17)(10,22)(11,18)(13,24)(14,16)(15,29)(19,23)(20,31)(21,34)(25,28)(26,36)(27,30)(32,33)(35,37);(6,9)(7,13)(8,17)(10,21)(11,20)(12,24)(14,27)(15,26)(16,30)(18,31)(19,32)(22,34)(23,33)(25,35)(28,37)(29,36)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5381);