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×#3514 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3153 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)(3,7)(9,10,14,17,13,18,12,16)(11,19,20,24,21,15,22,23);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,24)(17,23)(18,19);(9,11)(10,15)(12,20)(13,21)(14,22)(16,23)(17,24)(18,19);(2,5)(6,8)(9,12,13,14)(10,16,18,17)(11,20,21,22)(15,23,19,24);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,24,19,23)Polycyclic presentation:
2^9|a^2=h,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, 6293584);