Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
1959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,5)(3,7,8,6)(9,10,13,17)(11,19,21,16)(12,22,23,20)(14,24,15,18);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,14)(13,21)(15,23)(17,19)(18,22)(20,24);(2,5)(6,7)(9,12,15,11,13,23,14,21)(10,16,24,20,17,19,18,22);(1,3,4,8)(2,6,5,7)(10,17)(16,19)(18,24)(20,22);(9,10,13,17)(11,20,21,22)(12,16,23,19)(14,18,15,24)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fi,e^2=i,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,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592177);