Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#1794 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #120 |
| commutator | C2×C4 | C22×C4×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #379 |
| prime radical | C5×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ 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)(6,7,11,15,12,16,10,23)(8,17,19,13,20,28,24,27)(9,14,21,30,22,31,25,29)(18,33,36,37,35,32,34,26);(6,8)(7,13)(9,18)(10,24)(11,19)(12,20)(14,26)(15,28)(16,27)(17,23)(21,36)(22,35)(25,34)(29,32)(30,33)(31,37);(6,9)(7,14)(8,18)(10,25)(11,21)(12,22)(13,26)(15,30)(16,31)(17,32)(19,36)(20,35)(23,29)(24,34)(27,37)(28,33)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5384);