Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,16,23,14,13,15,17)(11,18,21,22,19,12,20,24);(1,2)(3,4)(6,7)(9,11,15,20,14,19,16,21)(10,12,17,22,13,18,23,24);(2,4)(5,6,8,7)(9,12)(10,11)(13,19)(14,18)(15,24)(16,22)(17,21)(20,23);(5,7,8,6)(9,13)(10,14)(11,18)(12,19)(15,23)(16,17)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=ei,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,f^c=fi,g^c=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 329695);