Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#106 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | C4×#106 |
| prime radical | C23 | C4×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
251 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7,4,6,8,3)(9,10,11,17,14,21,12,18,15,19,23,16,13,20,22,24);(2,6)(3,7)(10,16,19,17)(11,12)(13,14)(18,20,24,21)(22,23);(1,3,8,6,4,7,5,2)(9,11,14,12,15,23,13,22)(10,17,21,18,19,16,20,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=deg,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,f^a=fi,c^b=ch,d^b=dg,e^b=egi,g^b=gi,h^b=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225334, #72227102, #72227103, … |
| maximal normal subgroup of | #72225334, #72227102, #72227103, … |
| maximal quotient of | #72225325, #72227184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36356);