Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8465 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#379 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3272 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,15,17,20)(16,23,24,22);(2,5)(3,7)(9,10)(11,16)(12,15)(13,20)(14,17)(18,22)(19,24)(21,23);(2,5)(3,7)(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,24)(20,23);(9,12,14,13)(10,15,17,20)(11,18,19,21)(16,22,24,23)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fi,g^b=ghi,e^c=ei,g^c=gh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94234);