Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135678 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1835 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #8788 |
| prime radical | C23 | #430 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#97 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,13,9,15,5,14,10,16,6,26,8,27,25,31)(3,17,18,29,20,32,4,22,21,12,23,30,19,28,24,11);(1,3,5,24,8,19,7,23,10,21,25,4,9,20,6,18)(2,11,26,17,15,29,31,32,16,22,13,12,27,30,14,28);(1,4,5,19,8,18,7,21,10,24,25,20,9,23,6,3)(2,12,13,30,15,28,14,11,16,17,26,29,27,32,31,22)Polycyclic presentation:
2^9|a^2=f,b^2=eg,c^2=d,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,d^b=di,f^b=fi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42874);