Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1663 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
172 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 172 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) = 640 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,12,16)(8,17,19,20)(9,18,13,21)(10,15,11,14)(22,23,27,25)(24,30,33,37)(26,34,36,29)(28,35,31,32);(6,8)(7,13)(9,16)(10,21)(11,18)(12,19)(14,20)(15,17)(22,24,27,33)(23,28,25,31)(26,35,36,32)(29,37,34,30);(6,9,12,13)(7,8,16,19)(10,20,11,17)(14,18,15,21)(22,25,27,23)(24,31,33,28)(26,29,36,34)(30,35,37,32)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5253);