Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
484 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 484 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,3,4,6,8,7)(17,18)(19,20)(21,23)(22,24);(2,6)(3,7)(9,10,11,13,12,14,15,16)(17,19,21,24,22,23,20,18);(1,3,5,6,4,7,8,2)(17,20,22,21)(18,19,24,23)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=dg,f^a=fg,c^b=cgh152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3899);