Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150413 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2128 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | #407 | C23 |
| socle | C2 | #11754 |
| prime radical | #2010 | C22 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C42 ◁ #2920 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
349 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 349 conjugacy classes of subgroups — too many to draw. Only its 52 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 — 3 generators on 16 points:
(2,6)(3,10,12,14)(7,15)(8,11)(9,16);(1,2)(3,11)(4,7)(5,9)(6,14)(8,13)(10,16)(12,15);(2,7,9,8)(3,12)(6,15,16,11)(10,14)Polycyclic presentation:
2^9|a^2=e,c^2=h,d^2=gi,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,c^b=cf,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,e^d=ehi,f^d=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57609);