Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#2821 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| prime radical | C24×C5 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2,3,4,5)(10,11,13,15)(12,14,16,17)(19,23)(24,26)(25,29)(32,33);(7,9)(12,16)(14,17)(18,19)(20,26)(21,29)(22,23)(24,28)(25,31)(27,32)(30,33);(6,7)(8,9)(10,11)(12,14)(13,15)(16,17)(18,20,22,28)(19,24,23,26)(21,30,31,27)(25,33,29,32);(10,12)(11,14)(13,16)(15,17)(18,21,22,31)(19,25,23,29)(20,27,28,30)(24,32,26,33)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18498);