Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26,23,27,36,44)(19,28,31,39,30,43,21,24)(20,25,35,42,34,41,47,48)(29,45,33,40,32,46,37,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,30)(18,24,27,39)(20,32,34,29)(21,36,31,22)(25,40,41,38)(26,43,44,28)(33,47,37,35)(42,45,48,46);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,20,23,34)(18,25,27,41)(19,29,30,32)(21,37,31,33)(22,35,36,47)(24,38,39,40)(26,42,44,48)(28,45,43,46);(3,8)(5,7)(11,16)(13,15)(17,21,22,19,23,31,36,30)(18,24,26,28,27,39,44,43)(20,33,35,32,34,37,47,29)(25,40,42,46,41,38,48,45);(10,14)(13,15)(19,30)(20,34)(21,31)(24,39)(25,41)(28,43)(35,47)(42,48)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=h,d^2=fg,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6987811);