Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136861 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8973 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
358 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 358 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:
(1,2,6,12,7,10,8,3)(4,15,18,30)(5,11,22,29,23,20,24,14)(9,25,27,21)(13,17,26,32)(16,28,19,31);(1,3,6,10,7,12,8,2)(4,13,19,27,18,26,16,9)(5,20,22,29,23,11,24,14)(15,25,28,32,30,21,31,17);(1,4)(2,9)(3,13)(5,21)(6,19)(7,18)(8,16)(10,27)(11,28)(12,26)(14,30)(15,29)(17,22)(20,31)(23,25)(24,32)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,c^a=ce,d^a=di,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=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44057);