Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | C2×#95 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
674 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 674 conjugacy classes of subgroups — too many to draw. Only its 198 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:
(1,2,3,6,4,5,7,8)(9,10,12,14,13,15,11,16);(2,5)(3,7)(17,18,19,22,20,21,23,24);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(18,21)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ch,d^a=dg,c^b=ci,e^b=eg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2145);