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×#1660 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C5 |
| prime radical | C2×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
196 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 196 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) = 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)(14,15,20,24)(16,25,27,28)(17,26,21,29)(18,23,19,22);(8,12)(10,13)(14,16)(15,21)(17,24)(18,29)(19,26)(20,27)(22,28)(23,25);(6,7)(8,13)(9,11)(10,12)(14,17,20,21)(15,16,24,27)(18,28,19,25)(22,26,23,29)100 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5250);