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 | #31224 | C2 |
| LCS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,19,15,20,13,24)(11,21,22,17,23,16,12,18);(1,2)(3,6)(4,8)(5,7)(10,16)(11,22)(12,23)(13,14)(17,19)(18,24)(20,21);(1,3,5,4)(2,6,7,8)(10,17,20,18)(11,22)(12,23)(13,14)(16,19,21,24)Polycyclic presentation:
2^9|a^2=g,c^2=fg,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=dfi,e^b=egi,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=ei41 conjugacy classes — view the full character table.
| maximal subgroup of | #72265444, #72265457, #72265461, … |
| maximal normal subgroup of | #72265444, #72265457, #72265461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 33585);