Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#3127 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C23×C5 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,18,22)(16,24,26,20)(17,28,29,37)(19,32,31,23)(21,33,36,30)(25,40,41,45)(27,35,43,39)(34,42,38,44);(7,10)(11,13)(14,16)(15,20)(17,25)(18,26)(19,27)(21,34)(22,24)(23,35)(28,45)(29,41)(30,42)(31,43)(32,39)(33,44)(36,38)(37,40);(14,17,18,29)(15,21,22,36)(16,25,26,41)(19,33,31,30)(20,34,24,38)(23,37,32,28)(27,44,43,42)(35,40,39,45);(6,8,9,12)(7,11,10,13)(16,26)(20,24)(25,41)(27,43)(34,38)(35,39)(40,45)(42,44)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18804);