Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1443 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,23,28,24,29,22,27)(19,30,32,42,33,41,21,25)(20,26,36,44,37,45,39,46)(31,47,35,43,34,48,38,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,24,33)(18,25,29,42)(20,34,37,31)(21,22,32,23)(26,43,45,40)(27,30,28,41)(35,39,38,36)(44,47,46,48);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,24,37)(18,26,29,45)(19,31,33,34)(21,38,32,35)(22,39,23,36)(25,40,42,43)(27,46,28,44)(30,47,41,48);(3,8)(5,7)(11,16)(13,15)(17,21,23,19,24,32,22,33)(18,25,28,30,29,42,27,41)(20,35,36,34,37,38,39,31)(26,43,44,48,45,40,46,47);(10,14)(13,15)(17,22,24,23)(18,27,29,28)(19,21,33,32)(20,36,37,39)(25,41,42,30)(26,44,45,46)(31,35,34,38)(40,47,43,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6986756);