Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94453 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7318 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, prime radical, cosocle | #3490 | C22 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ #3490 ◁ G | |
| LECS | G ▷ #3490 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
362 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 38 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)(3,6,10,13)(5,9,12,16)(7,14,15,11);(2,6,9,7)(8,13)(10,14)(15,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=h,b^a=bc,c^a=cf,e^a=egh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfh,f^b=fh,f^c=fi,e^d=eh,g^d=gi32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#94453, #72225905, #72225907 |
| maximal normal subgroup of | C2×#94453, #72225905, #72225907 |
| maximal quotient of | #72213602, #72225886, C2×#94453, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1649);