Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#1307 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #95 |
| commutator | C23 | C42×C5 |
| Frattini, cosocle | C2×C42 | C22×C5 |
| prime radical | C23×C5 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,7,10,8,9,12,11,13)(14,15)(16,22,23,19)(17,21)(18,20)(24,28,29,26)(25,27);(6,8,11,12,9,13,10,7)(14,16,17,24,18,23,25,29)(15,19,21,26,20,22,27,28)130 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 440);