Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C4 |
| direct product | C5×#2164 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C23 |
| commutator | C22 | C22×C5×C8 |
| Frattini, cosocle | C2×C8 | C23×C5 |
| socle | C22×C5 | C22×C8 |
| prime radical | C23×C5 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
138 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 138 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,17,21,18,22,27,33,19,23,28,34,29,35,39,43)(16,20,25,31,26,32,38,42,24,30,36,40,37,41,44,45);(7,10)(11,13)(14,16)(15,20)(17,25)(18,26)(19,24)(21,31)(22,32)(23,30)(27,38)(28,36)(29,37)(33,42)(34,40)(35,41)(39,44)(43,45);(6,8,9,12)(7,11,10,13)(16,24)(20,30)(25,36)(26,37)(31,40)(32,41)(38,44)(42,45)280 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5754);