Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1217 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,33)(21,36)(22,28)(23,29)(24,27)(25,32)(26,35)(31,46)(34,44)(37,48)(38,42)(39,45)(40,43)(41,47);(9,11,12,16)(10,13,14,15)(17,19,23,32)(18,25,29,30)(20,24,35,21)(22,38,39,34)(26,27,33,36)(28,44,45,42)(31,40,47,37)(41,43,46,48);(1,3,4,8)(2,5,6,7)(10,14)(11,16)(17,20,24,32,23,35,21,19)(18,26,27,30,29,33,36,25)(22,34,40,47,39,38,37,31)(28,42,43,46,45,44,48,41);(10,14)(13,15)(17,21,23,24)(18,27,29,36)(19,20,32,35)(22,37,39,40)(25,33,30,26)(28,43,45,48)(31,34,47,38)(41,44,46,42);(17,22,21,37,23,39,24,40)(18,28,36,48,29,45,27,43)(19,31,35,38,32,47,20,34)(25,41,33,44,30,46,26,42)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=ghi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6725510);