Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500482 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,15,18)(11,13,17,14)(12,20,23,19)(16,22,24,21);(1,2)(3,7)(4,6)(5,8)(9,11,14,10,15,17,13,18)(12,20,21,24,23,19,22,16);(1,3)(2,5)(4,8)(6,7)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,20)(18,24);(3,8)(5,7)(9,13,15,14)(10,17,18,11)(12,21,23,22)(16,19,24,20);(2,6)(5,7)(9,13,15,14)(10,17,18,11)(12,22,23,21)(16,20,24,19)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407684);