Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9239 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
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 | |
349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10,13,18,15,19,23,22,14,17,20,24,12,11,16,21);(1,3,6,2)(4,7,5,8)(9,11,20,21,15,10,16,18,14,19,13,22,12,17,23,24);(1,2,6,3)(4,8,5,7)(9,12,14,15)(10,11,17,19)(13,16,20,23)(18,21,24,22)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=g,d^2=gi,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43913);