Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60536 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1918 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(6,8)(9,10,13,17,14,12,21,16)(11,15,19,23,20,18,24,22);(1,2)(3,6)(4,5)(7,8)(9,11,13,19,14,20,21,24)(10,15,17,23,12,18,16,22);(9,12)(10,14)(11,18)(13,16)(15,20)(17,21)(19,22)(23,24);(2,5)(6,8);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,g^c=gi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289574);