Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470832 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1460 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)(9,10,13,15)(11,16,14,12)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30);(2,5)(6,8)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31);(9,11,12,10,13,14,16,15)(19,25)(22,29)(26,31)(30,32);(1,3,4,7)(2,6,5,8)(19,25)(22,29)(26,31)(30,32);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)Polycyclic presentation:
2^9|a^2=hi,c^2=g,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6404942);