Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#2249 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C5 |
| prime radical | C2×C5×D16 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ G | |
184 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 184 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) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5)(7,9)(10,11)(12,20)(13,25)(14,17)(15,28)(16,22)(18,21)(19,26)(23,32)(24,39)(27,40)(29,38)(30,41)(31,36)(33,35)(34,37);(10,12,21,35,15,24,27,38,14,23,36,41,26,37,13,22)(11,16,25,34,19,30,31,32,17,29,40,39,28,33,18,20);(6,7,8,9)(11,17)(16,29)(18,31)(19,28)(20,32)(25,40)(30,33)(34,39)190 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5839);