Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437963 |
| 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 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,16,20)(11,14,19,15)(12,22,24,17)(13,18,23,21);(9,11,15,10,16,19,14,20)(12,18,23,17,24,21,13,22);(1,3,4,8)(2,5,6,7)(9,12,16,24)(10,17,20,22)(11,21,19,18)(13,14,23,15);(9,13)(10,18)(11,17)(12,14)(15,24)(16,23)(19,22)(20,21);(9,14,16,15)(10,11,20,19)(12,23,24,13)(17,21,22,18)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352992);