Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188766 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#117 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | C2×#432 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups; only the 182 normal subgroups are shown.
This group has 508 conjugacy classes of subgroups — too many to draw. Only its 182 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,19,12,18,21,16,14,17,23,24,22,15,11,20);(2,5)(3,7)(10,15,17,18)(11,21)(13,23)(16,24,20,19);(1,3,4,7)(2,6,5,8)(9,11,12,13,14,21,22,23)(10,16,18,24,17,20,15,19);(2,5)(6,8)(10,17)(15,18)(16,20)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=ceh,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403836);