Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8325 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #379 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,8,4,7,6,3)(9,10,13,16,14,17,12,18)(11,19,21,24,20,15,22,23);(1,3,6,7,4,8,5,2)(9,11)(10,15)(12,21)(13,22)(14,20)(16,24)(17,19)(18,23);(2,7)(3,8)(10,16)(11,20)(12,13)(15,23)(17,18)(19,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=ci,f^b=fh,e^c=eh,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12846);