Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437793 |
| 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 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2008 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)(3,6)(4,7)(5,8)(9,10,13,19,15,20,14,18)(11,16,22,21,12,17,24,23);(1,3,5,4)(2,6,8,7)(9,10)(11,21)(12,23)(13,18)(14,19)(15,20)(16,22)(17,24);(1,3,5,4)(2,7,8,6)(9,11)(10,16)(12,15)(13,22)(14,24)(17,20)(18,23)(19,21);(1,4,5,3)(2,7,8,6)(9,12)(10,17)(11,15)(13,24)(14,22)(16,20)(18,21)(19,23);(9,13,15,14)(10,18,20,19)(11,22,12,24)(16,23,17,21)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6360379);