Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#1634 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C5 |
| prime radical | C5×#555 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
218 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,19,22)(16,23,25,28)(17,21,18,26)(20,27,29,24);(8,12)(10,13)(14,16,19,25)(15,20,22,29)(17,24,18,27)(21,28,26,23);(6,7,9,11)(8,13,12,10)(14,17,19,18)(15,21,22,26)(16,24,25,27)(20,28,29,23)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5224);