Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | |
1231 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)(27,37)(28,42)(29,34)(30,35)(31,36)(32,40)(33,44)(38,46)(39,48)(41,47)(43,50)(45,49)(51,53)(52,56)(54,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,40,31,41,29,39)(26,32,34,37,36,48,35,47)(28,38,45,54,43,52,44,53)(33,46,50,51,49,56,42,55);(1,3,5,4)(2,7,8,6)(10,15)(11,17)(16,22)(18,23)(25,28,29,44,31,43,30,45)(26,33,35,42,36,49,34,50)(27,38,39,53,41,52,40,54)(32,46,47,55,48,56,37,51);(1,4,5,3)(2,6,8,7)(12,19)(16,22)(18,23)(20,24)(28,43)(33,49)(38,52)(42,50)(44,45)(46,56)(51,55)(53,54);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,29,31,30)(26,34,36,35)(27,39,41,40)(28,44,43,45)(32,37,48,47)(33,50,49,42)(38,53,52,54)(46,51,56,55)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=fi,d^2=fgh,e^2=hi,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=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6462278);