Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#476 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
539 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)(10,13)(11,16,20,23)(12,14)(17,18)(19,24,22,21);(1,3,4,7)(2,5,6,8)(9,10,12,13,15,18,14,17)(11,16,19,21,20,23,22,24);(2,6)(5,8);(9,11,12,19,15,20,14,22)(10,16,13,21,18,23,17,24)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,g^d=gh,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419549);