Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,14,12)(11,17,19,15)(13,20,21,16)(18,22,24,23)(25,26,30,35)(27,31,37,45)(28,40,42,48)(29,33,43,34)(32,47,49,41)(36,51,52,56)(38,46,39,50)(44,53,55,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,27,30,37)(26,31,35,45)(28,36,42,52)(29,39,43,38)(32,44,49,55)(33,50,34,46)(40,51,48,56)(41,54,47,53);(1,3)(2,5)(4,8)(6,7)(9,12,14,10)(11,17,19,15)(13,16,21,20)(18,22,24,23)(25,28,29,41,30,42,43,47)(26,32,34,40,35,49,33,48)(27,36,39,54,37,52,38,53)(31,44,46,51,45,55,50,56);(3,8)(5,7)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(27,37)(31,45)(36,52)(38,39)(44,55)(46,50)(51,56)(53,54);(2,6)(5,7)(11,19)(15,17)(18,24)(22,23)(25,29,30,43)(26,33,35,34)(27,38,37,39)(28,41,42,47)(31,46,45,50)(32,48,49,40)(36,53,52,54)(44,51,55,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818812);