Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | C4⋊Q8 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #18548 | C2 |
| LCS | G ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
486 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,6,11)(3,4,10,5)(7,12,15,16)(8,9,14,13);(1,3,5,9,8,12,7,2)(4,11,14,10,15,13,6,16);(2,9)(4,14)(5,7)(6,15)(10,11)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=df,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=egh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=egh,f^b=fi,h^b=hi,d^c=dg,e^c=eh,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72231252, #72231258, #72231263, … |
| maximal normal subgroup of | #72231252, #72231258, #72231263, … |
| maximal quotient of | #72280322, C2×#152606, #72338392, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59802);