Finite Group DatabasebetaFinite Group Databasebeta| order | 640 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#495 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C5 |
| prime radical | C22×C5×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
592 conjugacy classes of subgroups; only the 232 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(4,6)(7,8)(9,15)(10,18)(11,13)(12,16)(14,19)(17,21)(20,22);(7,9,10,16,11,17,19,22)(8,12,14,15,13,20,18,21);(3,4,5,6)(8,13)(12,20)(14,18)(15,21)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(640, 18823);