Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#388 |
| prime radical | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,12,17,14,18,13,15)(11,19,22,23,21,16,20,24);(1,2,5,7,6,3,4,8)(10,15,18,17)(16,23,19,24);(1,3,5,8,6,2,4,7)(9,11,13,20,14,21,12,22)(10,16,17,24,18,19,15,23);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,17,18,15)(11,20,21,22)(16,23,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=di,d^c=dh,f^c=fhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110368);