Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#3415 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C5 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C5×#41 ◁ C5×#131 ◁ G | |
370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,13)(9,12)(10,23)(11,16)(14,20)(15,21)(17,27)(18,24)(19,33)(22,26)(25,34)(28,32)(29,31)(30,35)(36,37);(7,12)(9,20)(10,21)(13,24)(14,15)(16,26)(18,32)(19,31)(22,34)(23,25)(27,35)(28,29)(30,37)(33,36);(6,8)(7,13)(9,18)(10,19)(11,17)(12,24)(14,28)(15,29)(16,27)(20,32)(21,31)(22,30)(23,33)(25,36)(26,35)(34,37);(8,17)(13,27)(18,30)(19,31)(24,35)(28,36)(29,33)(32,37)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 19092);