Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | |
1293 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)(25,26,31,37)(27,33,41,50)(28,42,45,34)(29,35,32,38)(30,36,40,49)(39,55,56,48)(43,54,47,51)(44,53,46,52);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,41)(26,33,37,50)(28,43,45,47)(29,40,32,30)(34,51,42,54)(35,49,38,36)(39,44,56,46)(48,52,55,53);(2,5)(6,8)(25,28,29,44,31,45,32,46)(26,34,35,52,37,42,38,53)(27,39,30,47,41,56,40,43)(33,48,36,54,50,55,49,51);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,31,32)(26,35,37,38)(27,40,41,30)(28,44,45,46)(33,49,50,36)(34,52,42,53)(39,43,56,47)(48,51,55,54);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(25,30,32,27,31,40,29,41)(26,36,38,33,37,49,35,50)(28,43,46,56,45,47,44,39)(34,51,53,55,42,54,52,48)Polycyclic presentation:
2^9|a^2=fi,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,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6738480);