Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | |
1216 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,7,4,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,31,36)(27,37,39,49)(28,42,44,52)(29,34,30,35)(32,40,48,41)(33,45,51,46)(38,47,43,50)(53,55,56,54);(2,6)(7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,41,31,39,30,40)(26,32,35,37,36,48,34,49)(28,43,45,54,44,38,46,55)(33,50,52,56,51,47,42,53);(1,3,5,4)(2,7,6,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,45,31,44,30,46)(26,33,35,52,36,51,34,42)(27,38,41,55,39,43,40,54)(32,47,37,53,48,50,49,56);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,39)(28,44)(32,48)(33,51)(37,49)(40,41)(42,52)(45,46);(1,4,5,3)(2,8,6,7)(12,19)(15,22)(18,23)(21,24)(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,45,44,46)(32,37,48,49)(33,42,51,52)(38,54,43,55)(47,53,50,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=f,d^2=fg,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476464);