Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11796 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
782 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 782 conjugacy classes of subgroups — too many to draw. Only its 138 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:
(2,5)(3,7)(6,8)(9,10,12,17,14,18,13,16)(11,19,21,15,20,24,22,23);(1,2)(3,5,7,8)(4,6)(9,11,13,22,14,20,12,21)(10,15,17,24,18,23,16,19);(2,6)(5,8)(11,20)(15,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=gi,f^2=gi,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh,h^b=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35047);