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 | #491 | 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 ▷ #491 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
305 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 305 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:
(1,2,5,9,6,10,19,28)(3,8,13,24,11,25,20,27)(4,14,17,26,15,21,30,31)(7,18,22,12,23,29,32,16);(2,7)(3,11)(4,15)(8,21)(9,26)(10,23)(13,20)(14,25)(18,29)(22,24)(27,32)(28,31);(1,3,6,11)(2,8,10,25)(4,16,15,12)(5,13,19,20)(7,21,23,14)(9,24,28,27)(17,18,30,29)(22,31,32,26)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=h,e^2=i,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, 60696);