Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | #121 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10,11,14,12,15,16,13);(1,3,5,4)(2,6,7,8)(17,18)(19,23)(20,22)(21,24);(1,4,5,3)(2,6,7,8)(10,13)(11,16)(14,15)(18,22)(19,20)(23,24);(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=e,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150368);