Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1852 |
| prime radical | C23 | #451 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#446 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
282 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,12,17,15,19,14,29)(11,20,21,38,23,37,30,34)(13,25,24,36,28,35,27,18)(16,26,31,40,32,39,33,22);(3,7)(5,8)(9,11,12,21,15,23,14,30)(10,16,17,31,19,32,29,33)(13,26,24,40,28,39,27,22)(18,34,25,20,36,38,35,37);(1,2,4,6)(3,8,7,5)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,27,28,24)(16,31,32,33)(18,35,36,25)(20,34,37,38)(22,39,40,26)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dg,e^b=eh,g^b=gi,d^c=dh,e^c=ehi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43548);