Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #238043 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#393 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1686 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)(3,5)(4,6)(7,8)(9,10)(11,18,20,24)(12,17)(13,22)(14,16)(15,21,23,19);(1,3)(2,5)(4,7)(6,8)(11,19,20,21)(15,18,23,24);(5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,24)(18,22);(10,16)(15,23)(17,22)(18,24)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^c=fh,g^c=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145239);