Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3196 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(9,10)(11,15)(12,14)(13,16)(18,22)(20,23);(2,4)(5,7,8,6)(9,10,12,14)(11,13,16,15)(18,22)(19,24);(1,2)(3,4)(11,16)(13,15)(17,18)(19,23)(20,24)(21,22);(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,23,22,20);(9,10)(11,13)(12,14)(15,16)(17,20,21,23)(18,24,22,19)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 468678);