Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C2×C42 |
| direct product | C5×#2844 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C24×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,9,11)(8,10,12,13)(14,15,18,17)(16,19,20,21);(8,12)(10,13)(15,17)(19,21)(22,23,25,27)(24,28,29,26);(6,7)(8,10)(9,11)(12,13)(14,16)(15,19)(17,21)(18,20)(22,24,25,29)(23,26,27,28);(6,8)(7,10)(9,12)(11,13)(14,17,18,15)(16,19,20,21)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18521);