Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #125892 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1629 |
| commutator, Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #453 |
| LCS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
312 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(5,8)(9,10)(11,19)(12,24,27,40)(13,18)(14,29)(15,31)(16,22)(17,26,36,21)(20,28)(23,34)(25,37,32,33)(30,38,39,35);(1,2)(3,5)(4,6)(7,8)(9,11,13,22)(10,16,18,19)(12,21,27,26)(14,28,23,15)(17,33,36,37)(20,29,31,34)(24,38,40,35)(25,39,32,30);(1,3,4,7)(2,5,6,8)(9,12,13,27)(10,17,18,36)(11,21,22,26)(14,30,23,39)(15,32,28,25)(16,33,19,37)(20,38,31,35)(24,29,40,34)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=f,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=eg,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33144);