Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
732 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 158 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:
(6,7)(9,10,13,18,14,19,11,15)(12,16,20,24,21,17,22,23);(1,2,3,4)(5,6,8,7)(9,11,14,13)(10,15,19,18)(12,20,21,22)(16,24,17,23);(5,7,8,6)(10,16)(11,13)(12,21)(15,23)(17,19)(18,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,e^2=i,f^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=ci,e^c=ei,f^c=fhi,f^e=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54063);