Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,3)(5,6,7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,23)(20,22)(21,24);(9,11)(10,14)(12,15)(13,16)(19,24)(21,23);(1,2,4,3)(17,19,20,24)(18,21,22,23);(1,3,4,2)(10,15)(11,16)(18,22)(21,23);(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 428044);