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 | |
1262 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)(25,26,29,34,31,36,30,35)(27,37,41,32,39,48,40,47)(28,42,44,50,45,49,43,33)(38,52,54,56,53,46,55,51);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27,31,39)(26,32,36,47)(28,38,45,53)(29,40,30,41)(33,46,50,52)(34,37,35,48)(42,56,49,51)(43,54,44,55);(10,14)(15,21)(16,22)(23,24)(25,28,31,45)(26,33,36,50)(27,38,39,53)(29,43,30,44)(32,46,47,52)(34,49,35,42)(37,51,48,56)(40,54,41,55);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(27,39)(32,47)(37,48)(38,53)(40,41)(46,52)(51,56)(54,55);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,43,45,44)(32,37,47,48)(33,49,50,42)(38,54,53,55)(46,51,52,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fgi,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,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, 6463210);