Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7734 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | #386 |
| prime radical | #2011 | C4 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
534 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,7,8)(10,13,14,16)(17,18,22,24,21,19,20,23);(1,2,6,5,4,7,3,8)(11,15)(13,16)(17,19,21,18)(20,23,22,24);(9,10)(11,16)(12,14)(13,15)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=hi,f^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=ehi,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897);