Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94547 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7345 |
| commutator | #125 | C2×C8 |
| Frattini, cosocle | #1994 | C22 |
| prime radical | #3548 | C4 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1994 ◁ G | |
| LECS | G ▷ #1994 ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
237 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 26 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:
(1,2,4,8,5,11,3,13,7,12,15,9,6,10,14,16);(2,8)(4,15)(5,6)(9,12)(10,13)(11,16)Polycyclic presentation:
2^9|a^2=d,c^2=h,d^2=f,e^2=h,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,d^b=de,e^b=eh,f^b=fi,g^b=ghi,d^c=dghi,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72272672 |
| maximal normal subgroup of | #72272672 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1744);