Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C23 | #386 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6,5,8,3,7)(9,10,12,15,13,11,14,16)(17,18,20,22,21,19,24,23);(2,6)(3,4)(7,8)(9,11,12,16,13,10,14,15)(17,19)(18,21)(20,22)(23,24);(2,6)(3,4)(7,8)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=egh,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eg,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6781);