Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 |
| prime radical | #60698 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1603 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,9)(3,10,12,7)(4,13,5,8)(11,15,14,16)(17,18)(19,22)(20,24)(21,27)(23,25)(26,30)(28,31)(29,32);(1,3,6,12)(2,7,9,10)(4,11,5,14)(8,15,13,16)(17,18)(19,21)(20,26)(22,27)(23,25)(24,30)(28,32)(29,31);(17,19,21,18,23,27,22,25)(20,26,29,24,28,32,30,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(20,28)(24,31)(26,32)(29,30);(1,5,6,4)(2,8,9,13)(3,11,12,14)(7,16,10,15)(17,20,22,30,23,28,21,29)(18,24,19,26,25,31,27,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=hi,d^2=f,e^2=fh,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,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6719189);