Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #126351 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator, Frattini, cosocle | C2×#97 | C23 |
| prime radical | C23 | #450 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
264 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,16,20,15,22,14,21)(11,23,31,39,27,38,26,32)(12,28,30,33,29,17,13,19)(18,34,35,25,37,24,36,40);(1,2)(3,8)(4,6)(5,7)(10,17,22,28)(12,13)(14,16)(18,32,37,39)(19,20,33,21)(23,35,38,36)(24,25)(26,31)(29,30)(34,40);(3,7)(5,8)(9,11,15,27)(10,18,22,37)(12,25,29,40)(13,24,30,34)(14,31,16,26)(17,32,28,39)(19,38,33,23)(20,36,21,35)Polycyclic presentation:
2^9|a^2=fgh,b^2=fgi,c^2=i,d^2=f,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,h^a=hi,d^b=df,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ehi,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33696);