Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907977 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1379 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,36,38,32)(28,40,43,52)(29,35,31,46)(33,45,51,42)(37,53,41,56)(39,49,44,48)(47,55,50,54);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,30,38)(26,32,34,36)(28,41,43,37)(29,44,31,39)(33,50,51,47)(35,49,46,48)(40,53,52,56)(42,54,45,55);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28,31,42,30,43,29,45)(26,33,35,40,34,51,46,52)(27,37,39,55,38,41,44,54)(32,47,49,56,36,50,48,53);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,34)(27,38)(33,51)(35,46)(37,41)(39,44)(40,52)(54,55);(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,34,46)(27,39,38,44)(28,42,43,45)(32,48,36,49)(33,52,51,40)(37,54,41,55)(47,56,50,53)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=g,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6831584);