Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12366 |
| 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 | |
356 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 356 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:
(2,7,10,14)(3,11,13,20,12,29,15,30)(5,18)(8,22,21,32,23,27,25,24)(9,26,28,19)(16,31);(1,2,4,14,6,10,17,7)(3,8,15,25,12,23,13,21)(5,19,16,9,18,26,31,28)(11,27,30,32,29,22,20,24);(1,3)(2,8)(4,15)(5,20)(6,12)(7,21)(9,27)(10,23)(11,16)(13,17)(14,25)(18,30)(19,24)(22,28)(26,32)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,h^b=hi,d^c=dh,e^c=eh,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, 44056);