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 | #8924 |
| prime radical | #3147 | C22 |
| 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 | |
350 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 88 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:
(1,2,6,12,8,4,7,11)(3,13,16,30,18,31,17,28)(5,20,23,19,25,22,24,10)(9,26,27,15,14,21,29,32);(1,3,7,16,8,18,6,17)(2,9,11,27,4,14,12,29)(5,21,24,32,25,26,23,15)(10,28,22,13,19,30,20,31);(1,4)(2,8)(3,14)(5,22)(6,11)(7,12)(9,18)(10,23)(13,21)(15,28)(16,29)(17,27)(19,24)(20,25)(26,31)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=fh,d^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dgi,e^b=eg,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43732);