Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,17)(12,21)(13,14)(15,19)(16,20)(18,23)(22,24);(1,3,5,4)(2,6,7,8)(9,11,14,21)(10,12,13,17)(15,16,23,22)(18,20,19,24);(9,12,14,17)(10,16,13,22)(11,19,21,18)(15,24,23,20);(1,4,5,3)(2,6,7,8)(9,11,14,21)(10,17,13,12)(15,16,23,22)(18,24,19,20);(9,13)(10,14)(11,20)(12,22)(15,19)(16,17)(18,23)(21,24)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484146);