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×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1562 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,23)(22,24)(25,29)(26,28)(27,30)(31,32);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,23)(19,25)(20,28)(22,31)(26,27)(29,30);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,19,21,27)(18,20,23,30)(22,26,31,25)(24,28,32,29);(1,4,5,3)(2,6,8,7)(18,23)(19,26)(20,29)(22,31)(25,27)(28,30);(17,20,21,30)(18,19,23,27)(22,28,31,29)(24,26,32,25)Polycyclic presentation:
2^9|b^2=fgi,c^2=f,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,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, 6488584);