Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#1395 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #99 |
| commutator | C2×C4 | C42×C5 |
| Frattini, cosocle | C2×#41 | C22×C5 |
| socle | C2×C5 | #342 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C2×C5×#41 ◁ G | |
162 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 162 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 — 2 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16,18,28)(9,20,21,13)(10,23)(11,15)(12,17)(14,24)(19,33,32,29)(22,27,35,34)(25,37)(26,31)(30,36);(6,8,10,19,11,18,24,32)(7,12,14,25,15,26,23,36)(9,17,22,30,21,31,35,37)(13,16,27,29,20,28,34,33)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 528);