Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2194 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| prime radical | C5×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#131 ◁ G | |
216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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)(6,7,12,18)(8,19,22,35)(9,23,27,34)(10,17,11,16)(13,29,32,20)(14,28,33,21)(15,25,30,26)(24,36,37,31);(6,8)(7,13)(9,24)(10,21)(11,28)(12,22)(14,30)(15,33)(16,31)(17,36)(18,32)(19,23)(20,26)(25,29)(27,37)(34,35);(6,8)(7,14)(9,20)(10,28)(11,21)(12,22)(13,30)(15,32)(16,35)(17,19)(18,33)(23,36)(24,26)(25,37)(27,29)(31,34)115 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5784);