Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #153493 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13569 |
| commutator | #125 | C22×C4 |
| Frattini, cosocle | #490 | C23 |
| prime radical | #19001 | C2 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #3142 ◁ G | |
| LECS | G ▷ #490 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,8)(4,14,17,7)(5,10)(6,9,18,26)(11,25)(12,28,29,23)(13,24,27,32)(15,21,19,22)(16,31,30,20);(2,7)(3,11)(4,15)(6,18)(8,20)(9,21)(10,22)(14,26)(23,25)(24,28)(29,30)(31,32);(1,3,5,11)(2,8,10,25)(4,16,15,12)(6,13,18,27)(7,20,22,23)(9,24,26,32)(14,31,21,28)(17,30,19,29)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=h,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=efi,f^b=fhi,g^b=gh,d^c=di,e^d=egh,f^d=fh,g^d=gi,f^e=fhi,g^e=gi29 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60690);