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 | #45094 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1622 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,23,28,21,24)(19,29,32,39,30,25,37,41)(20,33,34,40,36,26,38,44)(31,42,46,48,35,43,45,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,34)(21,22)(25,41)(27,28)(29,39)(31,45)(35,46)(36,38)(40,44)(42,43);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19,23,30)(18,25,28,29)(20,35,36,31)(21,37,22,32)(24,39,27,41)(26,43,33,42)(34,45,38,46)(40,48,44,47);(3,8)(5,7)(11,16)(13,15)(17,20,21,38,23,36,22,34)(18,26,24,40,28,33,27,44)(19,31,37,45,30,35,32,46)(25,42,39,47,29,43,41,48);(10,14)(13,15)(17,21,23,22)(18,27,28,24)(19,32,30,37)(20,34,36,38)(25,39,29,41)(26,40,33,44)(31,45,35,46)(42,48,43,47)Polycyclic presentation:
2^9|a^2=fgi,c^2=h,d^2=fg,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,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, 6987948);