Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12364 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1629 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8973 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
360 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,8,11,12)(3,13,14,30,15,31,21,18)(9,24,26,25,27,28,22,32)(10,23,29,17);(1,2,4,12,6,11,7,8)(3,9,14,26,15,27,21,22)(5,17,16,29,19,23,20,10)(13,24,30,25,31,28,18,32);(1,3)(2,9)(4,14)(5,18)(6,15)(7,21)(8,22)(10,28)(11,27)(12,26)(13,16)(17,32)(19,30)(20,31)(23,25)(24,29)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=h,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44054);