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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,7,4,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,36)(27,37,39,49)(28,42,46,51)(29,34,30,35)(32,41,48,40)(33,44,52,45)(38,47,43,50)(53,54,56,55);(2,6)(7,8)(9,11,13,17)(10,14,15,21)(12,18,19,23)(16,20,22,24)(25,27,30,41,31,39,29,40)(26,32,34,49,36,48,35,37)(28,43,45,54,46,38,44,55)(33,50,42,53,52,47,51,56);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,22)(18,23)(25,28,29,44,31,46,30,45)(26,33,35,51,36,52,34,42)(27,38,40,54,39,43,41,55)(32,47,37,53,48,50,49,56);(1,4,5,3)(2,8,6,7)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(27,39)(32,48)(37,49)(38,43)(40,41)(47,50)(53,56)(54,55);(12,19)(16,22)(18,23)(20,24)(25,29,31,30)(26,34,36,35)(27,40,39,41)(28,44,46,45)(32,49,48,37)(33,42,52,51)(38,54,43,55)(47,56,50,53)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=f,d^2=fg,e^2=i,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^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476192);