Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | C4×C22⋊C4 |
| prime radical | C22×D4 | C42 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(9,10)(11,16)(12,14)(13,15)(17,18,20,19,21,22,24,23);(1,2,3,6,4,5,7,8)(9,11,12,10,13,14,16,15);(2,5)(6,8)(17,19,20,22,21,23,24,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4044);