Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #138776 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| prime radical | C2×D42 | C22 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×D42 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
799 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 799 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,4,6,5,8,3,7)(9,10)(11,12,15,13)(14,16);(2,6)(3,4)(7,8)(9,11,12,16)(10,14,15,13);(2,6)(3,4)(7,8)(9,12)(10,15)(11,16)(13,14)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=gh,e^2=ghi,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=egh,f^b=fh,g^b=gi,h^b=hi,d^c=dgh,e^c=eghi,f^c=fgh,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72281272, #72281359, #72281537, … |
| maximal normal subgroup of | #72281272, #72281359, #72281537, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46088);