Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C2×C8 | C4×C8 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C22 | #1279 |
| prime radical | C23 | #339 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
242 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 64 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,13,17,15,18,23,24,16,19,22,21,14,11,12,20);(1,2,4,7,5,8,3,6)(9,11,14,19,16,18,15,10)(12,21,22,24,23,17,13,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=h,e^2=fg,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 656);