Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13);(2,6)(7,8)(17,18,23,27)(19,28,30,31)(20,29,24,25)(21,26,22,32);(9,11,12,16)(10,13,14,15)(17,19,23,30)(18,24,27,20)(21,29,22,25)(26,31,32,28);(1,3,5,4)(2,7,6,8)(17,20,23,24)(18,25,27,29)(19,26,30,32)(21,31,22,28);(1,4,5,3)(2,8,6,7)(17,21)(19,29)(22,23)(25,30)(26,32)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493725);