Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(9,10,13,15)(11,14,12,16)(17,18,21,24,20,22,19,23);(1,2,5,6)(3,8,4,7)(9,10)(11,16)(12,14)(13,15)(18,22)(23,24);(1,3,5,4)(2,7,6,8);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,19,20,21)(18,23,22,24);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19,20,21)(18,24,22,23)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316153);