Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8674 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #380 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1797 |
| prime radical | C23 | #407 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,4,3,7)(9,10,14,16,13,17,21,24,15,19,22,23,12,18,11,20);(1,3,6,5)(2,7,4,8)(9,11,12,22,15,21,13,14)(10,16,17,24,19,23,18,20);(1,4,6,2)(3,8,5,7)(9,12,15,13)(10,17,19,18)(11,14,21,22)(16,20,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=dg,c^2=g,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=ci,e^c=egi,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41598);