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 | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,15)(13,16)(18,23)(19,25)(26,32)(28,29);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,16,15)(17,18,21,23)(19,27,25,30)(20,29,24,28)(22,32,31,26);(1,3,5,4)(2,7,8,6)(9,11,13,12)(10,15,16,14)(17,19)(18,24)(20,23)(21,25)(22,28)(26,30)(27,32)(29,31);(9,12,13,11)(10,15,16,14)(18,23)(19,28)(20,30)(22,31)(24,27)(25,29);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,23)(21,24)(22,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=fg,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dh,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, 6498447);