Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213788 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1331 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)(3,6)(4,5)(7,8)(12,21)(15,22)(17,23)(20,24);(9,10)(11,17)(12,22)(13,16)(14,20)(15,21)(18,24)(19,23);(2,5)(6,8)(9,11)(10,14,16,18)(12,20)(13,19)(15,17,22,23)(21,24);(1,3,4,7)(2,6,5,8)(9,10)(11,18)(12,15)(13,16)(14,19)(17,24)(20,23)(21,22)Polycyclic presentation:
2^9|c^2=h,d^2=f,g^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290014);