Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#508 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | D8 |
| commutator, Frattini, cosocle | C8 | C24×C5 |
| socle | C22×C5 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C22×C5×C8 ◁ G | |
368 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 39 points:
(1,2)(35,36,37,38,39);(3,4)(5,14)(6,11)(7,19)(8,23)(9,13)(10,16)(12,21)(15,30)(17,31)(18,29)(20,25)(22,33)(24,28)(26,34)(27,32);(3,5)(4,10)(6,15)(7,20)(8,24)(9,17)(11,25)(12,28)(13,27)(14,19)(16,21)(18,32)(22,34)(23,26)(29,30)(31,33);(3,6,9,8)(4,11,13,23)(5,15,17,24)(7,18,22,21)(10,25,27,26)(12,19,29,33)(14,30,31,28)(16,20,32,34)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19088);