Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1445 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)(11,15)(14,16)(17,18,21,26,22,27,33,23)(19,24,30,38,29,37,43,34)(20,25,31,39,32,40,44,35)(28,36,41,47,42,46,48,45);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(18,23)(20,31)(21,33)(24,34)(26,27)(28,41)(30,43)(32,44)(35,39)(37,38)(42,48)(45,47);(2,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,34)(25,36)(26,38)(27,37)(31,41)(32,42)(33,43)(35,45)(39,47)(40,46)(44,48);(3,8)(5,7)(10,13)(14,16)(17,20,21,31,22,32,33,44)(18,25,26,39,27,40,23,35)(19,28,30,41,29,42,43,48)(24,36,38,47,37,46,34,45);(9,11,12,15)(10,14,13,16)(19,29)(24,37)(28,42)(30,43)(34,38)(36,46)(41,48)(45,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=fg,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7069475);