Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456417 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2424 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,5)(3,8,4,7)(9,10,13,16,14,12,21,17)(11,15,19,22,20,18,24,23);(1,3,6,4)(2,7,5,8)(9,11,13,19,14,20,21,24)(10,15,16,22,12,18,17,23);(1,4,6,3)(2,7,5,8)(9,12)(10,14)(11,15)(13,16)(17,21)(18,20)(19,23)(22,24);(1,5,6,2)(3,8,4,7)(9,12)(10,14)(11,18)(13,17)(15,20)(16,21)(19,23)(22,24)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=e,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364762);