Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1289 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(7,8)(9,10)(11,14)(12,20)(13,16)(15,19)(17,21)(18,22)(23,24);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,20)(16,22)(17,23)(21,24);(1,3,5,4)(2,7,6,8)(12,21)(14,22)(17,20)(19,24);(1,4,5,3)(2,7,6,8)(9,11)(10,15,16,23)(12,19)(13,18)(14,20,22,17)(21,24)Polycyclic presentation:
2^9|b^2=e,c^2=e,d^2=eg,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 355525);