Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94447 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7310 |
| commutator | C22×D4 | C42 |
| Frattini, prime radical, cosocle | #3484 | C22 |
| LCS | G ▷ C22×D4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3484 ◁ G | |
| LECS | G ▷ #3484 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 441 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:
(1,2,3,7)(4,6,10,14)(5,9,11,15)(8,12,16,13);(2,6)(3,10,11,12)(8,9)(14,16)Polycyclic presentation:
2^9|a^2=d,b^2=e,f^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfi,f^b=fh,d^c=dhi,f^c=fi,e^d=ehi,f^d=fi,g^d=gi32 conjugacy classes — view the full character table.
| maximal quotient of | #72266386 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1643);