Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(10,14)(12,19)(15,21)(17,20)(25,26,30,35,31,36,29,34)(27,32,40,48,38,46,39,47)(28,41,42,49,43,50,44,33)(37,51,53,55,52,56,54,45);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27,31,38)(26,32,36,46)(28,37,43,52)(29,39,30,40)(33,45,49,55)(34,47,35,48)(41,51,50,56)(42,53,44,54);(2,5)(3,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28,31,43)(26,33,36,49)(27,37,38,52)(29,42,30,44)(32,45,46,55)(34,41,35,50)(39,53,40,54)(47,51,48,56);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(27,38)(32,46)(37,52)(39,40)(45,55)(47,48)(51,56)(53,54);(2,5)(6,8)(10,14)(15,21)(16,22)(23,24)(25,29,31,30)(26,34,36,35)(27,39,38,40)(28,42,43,44)(32,47,46,48)(33,41,49,50)(37,53,52,54)(45,51,55,56)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=fi,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6844370);