Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| 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 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(6,7)(9,10,15,24,13,16,14,17)(11,18,21,22,19,12,20,23);(1,2)(3,4)(5,6,8,7)(9,11,14,20,13,19,15,21)(10,12,17,22,16,18,24,23);(2,4)(5,6,8,7)(9,12)(10,11)(13,18)(14,23)(15,22)(16,19)(17,21)(20,24);(5,7,8,6)(9,10)(11,12)(13,16)(14,17)(15,24)(18,19)(20,22)(21,23)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,f^c=fi,g^c=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 304895);