Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2080 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | #397 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ G | |
190 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 72 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)(8,16,19,29)(9,13)(10,14)(11,15)(12,18,26,23)(17,32,20,31)(21,27)(22,28)(24,30)(25,34,36,33)(35,37);(6,8)(7,12)(9,20)(10,23)(11,19)(13,25)(14,29)(15,26)(16,30)(17,22)(18,24)(21,33)(27,32)(28,36)(31,37)(34,35);(6,9,10,21,11,22,24,35)(7,13,14,27,15,28,30,37)(8,17,18,33,19,20,23,34)(12,25,16,31,26,36,29,32)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5670);