Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(10,14)(12,16)(18,20)(22,23);(2,4)(9,10,13,14)(11,12,15,16)(17,18,21,20)(19,23,24,22);(1,2)(3,4)(10,14)(11,15)(17,18,21,20)(19,23,24,22);(9,11,13,15)(10,12,14,16)(17,19,21,24)(18,22,20,23);(9,12)(10,15)(11,14)(13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451665);