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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1241 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)(11,14,17,21)(12,16,19,22)(18,20,23,24)(25,26,30,35)(27,36,40,47)(28,41,43,52)(29,33,37,34)(31,45,48,39)(32,49,51,44)(38,50,42,46)(53,56,55,54);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,37,45,30,40,29,39)(26,31,33,47,35,48,34,36)(28,42,49,54,43,38,44,56)(32,50,52,53,51,46,41,55);(1,3,5,4)(2,7,8,6)(10,15)(11,17)(16,22)(18,23)(25,28,37,49,30,43,29,44)(26,32,33,52,35,51,34,41)(27,38,45,56,40,42,39,54)(31,46,47,55,48,50,36,53);(1,4,5,3)(2,6,8,7)(12,19)(16,22)(18,23)(20,24)(28,43)(32,51)(38,42)(41,52)(44,49)(46,50)(53,55)(54,56);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,29,30,37)(26,33,35,34)(27,39,40,45)(28,44,43,49)(31,47,48,36)(32,52,51,41)(38,54,42,56)(46,55,50,53)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fgh,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6462294);