Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24425 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,12,13)(11,15,16,14)(17,18,22,27)(19,23,29,36)(20,31,33,40)(21,25,34,26)(24,39,41,32)(28,42,43,48)(30,37,45,38)(35,46,47,44);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,35)(25,37)(26,38)(27,36)(31,42)(32,44)(33,43)(34,45)(39,46)(40,48)(41,47);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,20,21,32,22,33,34,39)(18,24,26,31,27,41,25,40)(19,28,30,44,29,43,45,46)(23,35,38,42,36,47,37,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(19,29)(23,36)(28,43)(30,45)(35,47)(37,38)(42,48)(44,46);(2,6)(5,7)(17,21,22,34)(18,25,27,26)(19,30,29,45)(20,32,33,39)(23,37,36,38)(24,40,41,31)(28,44,43,46)(35,48,47,42)Polycyclic presentation:
2^9|a^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=efh,d^c=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818184);