Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #99 |
| commutator | C23 | C4×C16 |
| Frattini, cosocle | C2×C8×D4 | C22 |
| socle | C22 | #1282 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8×D4 ▷ C22×C4 ▷ C22 ▷ C1 | |
261 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 261 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,12,17,14,11,16,20,15,19,13,18,23,22,24,21);(2,5)(9,11,13,21,14,19,24,17,15,22,12,20,23,10,16,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=g,e^2=gh,g^2=i,b^a=bc,c^a=cf,f^a=fh,d^b=df,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 192);