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 | C22×D8 | 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 | |
296 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 296 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) = 128 |
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)(17,18,21,28,23,30,22,29)(19,27,31,25,32,24,20,26);(1,3,4,14,6,12,5,11)(2,7,13,10,9,16,8,15)(18,24,30,27)(19,20)(21,22)(25,29,26,28)(31,32);(18,25)(19,31)(20,32)(21,22)(24,29)(26,30)(27,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=h,f^2=ghi,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 | #72225273, #72226988, #72227161, … |
| maximal normal subgroup of | #72225273, #72226988, #72227161, … |
| maximal quotient of | #72225152 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43678);