Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,17,18,14)(12,16,20,22)(19,23,24,21)(25,26,30,34)(27,36,38,31)(28,40,43,50)(29,33,44,35)(32,48,51,42)(37,52,41,55)(39,47,54,46)(45,53,49,56);(1,3,5,4)(2,6,8,7)(9,11,13,18)(10,14,15,17)(12,19,20,24)(16,21,22,23)(25,27)(26,31)(28,41)(29,39)(30,38)(32,49)(33,46)(34,36)(35,47)(37,43)(40,52)(42,56)(44,54)(45,51)(48,53)(50,55);(1,3,5,4)(2,7,8,6)(10,15)(11,18)(16,22)(19,24)(25,28,29,42,30,43,44,48)(26,32,35,40,34,51,33,50)(27,37,39,53,38,41,54,56)(31,45,47,55,36,49,46,52);(1,4,5,3)(2,6,8,7)(12,20)(16,22)(19,24)(21,23)(27,38)(31,36)(37,41)(39,54)(45,49)(46,47)(52,55)(53,56);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)(25,29,30,44)(26,33,34,35)(27,39,38,54)(28,42,43,48)(31,46,36,47)(32,50,51,40)(37,53,41,56)(45,52,49,55)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fgi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6468016);