Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,14,16,15,18,13,12)(11,17,22,23,20,24,21,19);(5,7,8,6)(10,16)(12,18)(13,14)(17,23)(19,24)(21,22);(2,4)(9,11)(10,17)(12,19)(13,21)(14,22)(15,20)(16,23)(18,24);(9,12,14,10,15,16,13,18)(11,19,22,17,20,23,21,24);(11,20)(17,24)(19,23)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=fi,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289160);