Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3517 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,5)(3,7)(9,10,15,19,14,20,13,18)(11,17,21,24,12,16,22,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,12)(10,16,20,17)(13,22,15,21)(18,24,19,23);(9,12,14,11)(10,17,20,16)(13,22,15,21)(18,24,19,23);(2,5)(6,8)(9,13,14,15)(10,18,20,19)(11,21,12,22)(16,23,17,24);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,19,20,18)(11,21,12,22)(16,24,17,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293587);