Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#1722 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C5 |
| prime radical | C5×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,11)(12,13)(14,15,18,21)(16,22,23,29)(17,25,26,20)(19,27,28,24);(14,16,17,24,18,23,26,27)(15,19,20,22,21,28,25,29);(6,7,11,13,9,12,8,10)(16,23)(19,28)(22,29)(24,27)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 5312);