Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6632036 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,11,13)(12,14,16,15)(17,18)(19,25)(20,28)(21,24)(22,27)(23,30)(26,31)(29,32);(2,5)(3,7)(10,13)(14,15)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31);(1,3,4,7)(2,6,5,8)(10,14)(12,16)(13,15);(2,5)(6,8)(10,13)(14,15)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32);(18,24)(22,25)(23,28)(31,32)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=h,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efhi,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6558805);