Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1402 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)(11,13)(12,14)(15,16)(17,18,21,24)(19,23,25,20)(22,26,31,32)(27,29,28,30);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,24,20)(22,27,31,28)(26,29,32,30);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(18,24)(19,27)(20,29)(22,31)(23,30)(25,28);(1,4,5,3)(2,6,8,7)(17,20)(18,25)(19,24)(21,23)(22,30)(26,28)(27,32)(29,31);(19,27)(20,30)(22,31)(23,29)(25,28)(26,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,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, 6488765);