Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11757 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C22⋊C8 | C23 |
| socle | C22 | C2×#324 |
| prime radical | C24 | C2×M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C22⋊C8 ▷ C22×C4 ▷ C22 ▷ C1 | |
300 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,10,6,9,3,7)(4,8,12,15,13,16,11,14)(17,18,20,25,22,23,28,30,21,26,29,32,19,24,27,31);(1,3,6,5)(2,7)(4,12,13,11)(8,15)(9,10)(14,16)(18,23,26,24)(25,31,32,30);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,19,21,22)(18,24,26,23)(20,27,29,28)(25,31,32,30)Polycyclic presentation:
2^9|a^2=e,b^2=fh,c^2=h,d^2=i,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=chi,d^b=dhi,e^b=efi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 29553);