Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1518 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,21,24)(19,23,29,31)(20,25,30,28)(22,26,32,27);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,21,29)(18,23,24,31)(20,26,30,27)(22,25,32,28);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,24)(19,25)(20,23)(22,32)(28,29)(30,31);(17,20)(18,25)(19,27)(21,30)(22,23)(24,28)(26,29)(31,32);(19,28)(20,23)(22,32)(25,29)(26,27)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486814);