Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26,31,36)(27,37,38,51)(28,33,43,50)(29,35,30,34)(32,44,46,40)(39,53,41,56)(42,52,45,49)(47,54,48,55);(1,2,5,6)(3,8,4,7)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(27,38)(32,46)(37,51)(39,41)(40,44)(47,48)(53,56)(54,55);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,30,40,31,38,29,44)(26,32,35,37,36,46,34,51)(28,41,42,54,43,39,45,55)(33,48,49,56,50,47,52,53);(1,3,5,4)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28,31,43)(26,33,36,50)(27,39,38,41)(29,45,30,42)(32,47,46,48)(34,52,35,49)(37,53,51,56)(40,55,44,54);(1,4,5,3)(2,7,6,8)(10,16)(11,18)(15,24)(17,19)(25,29,31,30)(26,34,36,35)(27,40,38,44)(28,42,43,45)(32,37,46,51)(33,49,50,52)(39,54,41,55)(47,56,48,53)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6958234);