Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #136371 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#135 | Q8 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
269 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,7,12,15,14,16)(18,25,27,19)(20,21)(22,23)(26,28,30,29)(31,32);(1,3,4,14,6,12,5,11)(2,7,13,10,9,16,8,15)(17,18,22,29,24,27,23,28)(19,20,26,21,25,32,30,31);(17,19,24,25)(18,20,27,32)(21,28,31,29)(22,26,23,30)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=i,d^2=g,e^2=h,f^2=gh,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,d^b=dg,e^b=eh,f^b=fi,g^b=gi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72287131, #72287170, #72287207, … |
| maximal normal subgroup of | #72287131, #72287170, #72287207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43677);