Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
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 | C2×#127 | C2×C4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
436 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 436 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(2,5,6,8)(9,10,13,15,12,11,14,16)(17,18,20,23,22,19,21,24);(3,7)(5,8)(9,11,13,16,12,10,14,15)(17,19)(18,22)(20,23)(21,24);(1,2)(3,8)(4,6)(5,7)(10,11)(15,16)(17,19,22,18)(20,24,21,23)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6740);