Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C23 | C4×C16 |
| Frattini, cosocle | C22×C4×C8 | C22 |
| socle | C23 | #336 |
| prime radical | C24 | C4×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
266 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,11,12,15,14,16)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(1,3)(2,5,6,7)(4,8)(9,11,14,15,12,16,13,10)Polycyclic presentation:
2^9|a^2=d,b^2=e,d^2=h,e^2=f,h^2=i,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 690);