Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94516 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7338 |
| commutator | C23×C4 | C2×C8 |
| Frattini, cosocle | #1781 | C22 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
491 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 24 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) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,8)(3,7,11,15,10,6,12,14)(5,9,13,16);(2,6)(3,10)(7,9)(8,14)(11,12)(15,16)Polycyclic presentation:
2^9|a^2=d,c^2=i,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=dei,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72227526, #72227527, #72227528, … |
| maximal normal subgroup of | #72227526, #72227527, #72227528, … |
| maximal quotient of | #72227528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1712);