Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 32 points:
(2,6)(3,4)(7,8)(9,10,11,14,12,13,15,16)(17,18,20,24,21,25,29,32,22,26,19,23,27,30,28,31);(1,2)(3,6)(4,8)(5,7)(10,13)(11,15);(1,3,5,4)(2,6,7,8)(17,19,21,28,22,20,27,29)(18,23,25,31,26,24,30,32)Polycyclic presentation:
2^9|a^2=e,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,d^b=dg,e^b=efg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31812);