Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1212 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,29,33,30,34,43,49)(27,35,37,45,38,48,42,31)(28,39,41,55,40,32,46,47)(36,51,53,44,52,50,56,54);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,27,30,38)(26,31,34,45)(28,36,40,52)(29,42,43,37)(32,44,39,54)(33,48,49,35)(41,56,46,53)(47,51,55,50);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(25,28,30,40)(26,32,34,39)(27,36,38,52)(29,41,43,46)(31,44,45,54)(33,47,49,55)(35,50,48,51)(37,53,42,56);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)(28,40)(32,39)(36,52)(41,46)(44,54)(47,55)(50,51)(53,56)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=fi,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855435);