Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1237 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,12)(17,18)(19,30)(20,26)(21,39)(22,28)(23,29)(24,27)(25,33)(31,45)(32,41)(34,47)(35,48)(36,42)(37,44)(38,43)(40,46);(9,10)(11,16)(12,15)(13,14)(17,19,23,33)(18,25,29,30)(20,34,37,36)(21,32,24,22)(26,42,44,47)(27,41,39,28)(31,35,46,38)(40,48,45,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,24,38,23,37,21,35)(18,26,27,43,29,44,39,48)(19,31,32,34,33,46,22,36)(25,40,28,42,30,45,41,47);(9,12,13,11)(10,15,14,16)(17,21,23,24)(18,27,29,39)(19,32,33,22)(20,35,37,38)(25,41,30,28)(26,43,44,48)(31,34,46,36)(40,47,45,42);(1,3,4,7)(2,6,5,8)(17,22,21,33,23,32,24,19)(18,28,39,25,29,41,27,30)(20,36,35,46,37,34,38,31)(26,42,48,40,44,47,43,45)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=h,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6740682);