Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232393 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,12)(15,16)(17,18)(19,21)(20,22)(23,24);(1,3,4,8)(2,5,6,7)(19,24)(21,23);(9,10,13,16)(11,14,12,15)(18,21)(19,24)(22,23);(9,11,13,12)(10,15,16,14)(18,22)(21,23)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=hi,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dhi,f^c=fh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145041);