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 | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,8,7)(10,12)(14,15)(18,21)(22,24);(2,4)(9,10,13,12)(11,15,16,14)(18,21)(19,23);(1,2,3,4)(5,7,8,6)(9,10,13,12)(11,15,16,14)(17,18,20,21)(19,22,23,24);(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,23)(21,24);(9,12,13,10)(11,15,16,14)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fhi,d^2=f,e^2=f,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, 452693);