Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342866 |
| 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×#3546 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3243 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)(9,10,15,20,14,19,13,18)(11,21,22,12,17,16,24,23);(1,2)(3,6)(4,5)(7,8)(9,11,14,17)(10,16,19,21)(12,20,23,18)(13,24,15,22);(9,12,15,21,14,23,13,16)(10,17,18,24,19,11,20,22);(2,5)(6,8)(9,13,14,15)(10,18,19,20)(11,22,17,24)(12,16,23,21);(1,3,4,7)(2,6,5,8)(10,19)(11,17)(18,20)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301066);