Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184811 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1194 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)(3,7)(4,5)(6,8)(9,10,12,15,13,14,16,11)(17,18,21,23,22,19,20,24);(2,5)(6,7)(9,11,12,14,13,15,16,10)(17,19,21,24,22,18,20,23);(1,3,4,8)(2,6,5,7);(10,14)(11,15)(17,20,22,21)(18,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=f,d^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=di,e^b=eh,g^b=gh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 114123);