Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533105 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #25631 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1800 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,20,23,19,21,24,22);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(18,21)(22,23);(10,11)(14,15)(18,22)(20,24)(21,23);(1,4,5,3)(2,6,7,8)(12,16)(14,15)(18,21)(22,23);(9,12,13,16)(10,14,11,15)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6456224);