Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500157 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(9,10,12,17,14,19,13,18)(11,20,22,16,23,24,21,15);(1,2)(3,7)(4,6)(5,8)(9,11,14,23)(10,15,19,16)(12,21,13,22)(17,24,18,20);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,21)(14,23)(15,19)(17,24)(18,20);(3,8)(5,7)(9,12,14,13)(10,17,19,18)(11,21,23,22)(15,24,16,20);(2,6)(5,7)(9,12,14,13)(10,18,19,17)(11,22,23,21)(15,24,16,20)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,e^b=eh,f^b=fi,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, 6407361);