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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,37,39,32)(28,33,42,50)(29,35,44,46)(31,36,41,48)(38,52,54,45)(40,49,55,47)(43,51,53,56);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,39)(26,32,34,37)(28,40,42,55)(29,31,44,41)(33,47,50,49)(35,48,46,36)(38,53,54,43)(45,56,52,51);(2,5)(6,8)(25,28,41,53,30,42,31,43)(26,33,48,56,34,50,36,51)(27,38,44,55,39,54,29,40)(32,45,35,49,37,52,46,47);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,34)(28,42)(32,37)(35,46)(36,48)(38,54)(40,55)(43,53);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,29,41,27,30,44,31,39)(26,35,48,37,34,46,36,32)(28,40,53,38,42,55,43,54)(33,49,56,52,50,47,51,45)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=hi,d^2=fi,e^2=ghi,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, 6738368);