Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682628 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1961 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,15,18)(11,16,21,24)(12,19,20,23)(13,22,14,17);(1,2)(3,6)(4,8)(5,7)(9,11,15,21)(10,16,18,24)(12,13,20,14)(17,19,22,23);(9,12,13,11,15,20,14,21)(10,16,17,19,18,24,22,23);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,10,15,18)(11,19,21,23)(12,16,20,24)(13,17,14,22)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=g,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594177);