Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9443 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #386 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1869 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,9)(4,12,14,7)(5,11)(6,8)(10,13,16,15)(17,18,21,26,23,27,31,30,22,25,28,32,20,19,24,29);(1,3,9,2,6,11,5,8)(4,10,13,7,14,16,15,12)(17,19,28,29,23,18,24,26,22,27,21,30,20,25,31,32);(1,4,6,14)(2,7,8,12)(3,10,11,16)(5,15,9,13)(17,20,22,23)(18,19,25,27)(21,24,28,31)(26,29,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=di,e^b=egi,f^b=fi,d^c=dgi,f^c=fgi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 45111);