Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#2739 |
| 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 | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,10)(11,13)(14,15,17,19)(16,20,21,18)(22,23)(24,29)(25,26)(27,28);(7,10)(11,12)(14,15,17,19)(16,18,21,20)(22,24,27,25)(23,26,28,29);(6,8,9,13)(7,11,10,12)(14,15)(16,18)(17,19)(20,21)(22,25,27,24)(23,26,28,29);(6,7)(8,11)(9,10)(12,13)(14,16)(15,18)(17,21)(19,20)(22,26)(23,24)(25,28)(27,29)160 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18416);