Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94449 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7313 |
| commutator | #117 | C42 |
| Frattini, prime radical, cosocle | #3508 | C22 |
| LCS | G ▷ #117 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #3508 ◁ G | |
| LECS | G ▷ #3508 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 306 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)(4,12,13,15)(5,9,11,16)(6,14,8,10);(1,2)(3,6)(4,12,14,8)(5,9)(7,10,16,13)(11,15)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,f^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=dfh,f^b=fh,h^b=hi,d^c=dhi,g^c=gi,e^d=eh,f^d=fi,g^d=gi,f^e=fi32 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1645);