Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C24 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#323 |
| prime radical | C26 | C8 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
1612 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1612 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(2,5)(6,7);(10,14)(15,20)(16,21)(22,24)Polycyclic presentation:
2^9|a^2=e,e^2=g,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,e^b=ef,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29424);