Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1261 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(25,26,29,34,30,35,31,36)(27,37,40,48,39,32,41,47)(28,42,45,33,44,50,43,49)(38,52,55,51,53,56,54,46);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27,30,39)(26,32,35,37)(28,38,44,53)(29,40,31,41)(33,46,49,51)(34,47,36,48)(42,56,50,52)(43,54,45,55);(10,14)(15,21)(16,22)(23,24)(25,28,30,44)(26,33,35,49)(27,38,39,53)(29,43,31,45)(32,46,37,51)(34,42,36,50)(40,54,41,55)(47,56,48,52);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(27,39)(32,37)(38,53)(40,41)(46,51)(47,48)(52,56)(54,55);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,30,31)(26,34,35,36)(27,40,39,41)(28,43,44,45)(32,47,37,48)(33,42,49,50)(38,54,53,55)(46,56,51,52)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=i,d^2=h,e^2=hi,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6456106);