Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator, prime radical | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1218 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)(9,10,13,16,14,15,12,11)(17,18,22,23,21,24,20,19);(2,5)(6,7)(9,11,13,15,14,16,12,10)(17,19,20,24,21,23,22,18);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,11,15,16)(17,20,21,22)(18,23,24,19);(17,20,21,22)(18,19,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=efg,c^2=fh,d^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,e^b=eh,g^b=gh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 114078);