Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1294 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)(17,18,21,19)(20,24,29,23)(22,25,30,26)(27,31,32,28);(2,6)(7,8)(17,19,21,18)(20,27,29,32)(22,25,30,26)(23,28,24,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,21,29)(18,23,19,24)(22,31,30,28)(25,27,26,32);(1,4,5,3)(2,8,6,7)(18,19)(20,28)(22,30)(23,27)(24,32)(29,31);(17,20,21,29)(18,24,19,23)(22,31,30,28)(25,32,26,27)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493544);