Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#2212 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| prime radical | C5×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×C8 ◁ G | |
154 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 60 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)(9,21)(10,25)(11,27)(12,17)(13,19)(14,18)(15,26)(20,34)(22,33)(23,37)(24,29)(28,30)(31,35)(32,36);(6,8,10,19,12,20,26,11)(7,13,15,16,17,27,25,34)(9,18,22,35,24,36,23,28)(14,29,30,33,32,21,31,37);(6,9)(7,14)(8,18)(10,22)(11,28)(12,24)(13,29)(15,30)(16,33)(17,32)(19,35)(20,36)(21,27)(23,26)(25,31)(34,37)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5802);