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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,33)(28,42)(29,38)(30,39)(32,41)(34,45)(35,40)(37,44)(43,48)(46,47);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,35,22,30)(25,37,41,47)(26,40,36,39)(31,42,45,48)(32,46,33,44);(1,2)(3,8)(4,6)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,33,23,34,21,32)(18,25,26,31,27,41,36,45)(19,28,30,44,29,43,35,46)(24,37,40,42,38,47,39,48);(1,3,5,4)(2,6,7,8)(11,16)(13,15)(19,29)(24,38)(28,43)(30,35)(37,47)(39,40)(42,48)(44,46);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,29,35)(20,32,34,33)(24,39,38,40)(25,31,41,45)(28,44,43,46)(37,48,47,42)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^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=dfh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6822586);