Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#1757 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | C22⋊C4 |
| commutator | C22 | C2×C42×C5 |
| Frattini, cosocle | C22×C4 | C23×C5 |
| socle | C2×C5 | C4×C22⋊C4 |
| prime radical | C2×C5×#41 | C22 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
284 conjugacy classes of subgroups; only the 148 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 148 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)(6,7,10,14,11,15,22,28)(8,16,19,30,18,12,23,25)(9,13,20,26,21,27,33,36)(17,29,32,37,31,24,34,35);(6,8)(7,12,15,16)(9,17)(10,19)(11,18)(13,24,27,29)(14,25,28,30)(20,32)(21,31)(22,23)(26,35,36,37)(33,34);(6,9,10,20,11,21,22,33)(7,13,14,26,15,27,28,36)(8,17,19,32,18,31,23,34)(12,24,25,35,16,29,30,37)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5347);