Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395238 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,20)(13,19)(15,17)(16,23)(18,21)(22,24);(1,3,5,4)(2,6,8,7)(9,10)(11,13)(12,16)(14,19)(15,17)(18,21)(20,23)(22,24);(1,3,5,4)(2,7,8,6)(9,11,14,10,15,19,13,17)(12,18,22,16,23,24,21,20);(1,4,5,3)(2,6,8,7)(9,12,13,21,15,23,14,22)(10,16,11,18,17,20,19,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=efi,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308663);