Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1595 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)(19,28,23,24)(20,31,33,39)(21,35,38,26)(25,29,42,34)(30,40,36,43)(32,48,37,47)(41,46,44,45);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,23)(18,24,27,28)(20,29,33,34)(21,36,38,30)(25,39,42,31)(26,43,35,40)(32,46,37,45)(41,47,44,48);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,20,19,29,22,33,23,34)(18,25,24,39,27,42,28,31)(21,37,30,46,38,32,36,45)(26,44,40,47,35,41,43,48);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,30)(20,32)(22,38)(23,36)(24,40)(25,41)(27,35)(28,43)(29,45)(31,47)(33,37)(34,46)(39,48)(42,44);(2,6)(5,7)(21,38)(26,35)(30,36)(32,37)(40,43)(41,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=fi,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=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818383);