Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #504125 |
| 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 | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
446 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)(3,7)(4,5)(6,8)(9,10,14,17,15,13,12,18)(11,16,23,24,21,22,19,20);(2,5)(6,7)(9,11,12,19,15,21,14,23)(10,16,18,20,13,22,17,24);(10,17)(11,16,21,22)(12,14)(13,18)(19,24,23,20);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=fh,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 411430);