Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1813 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,32,27,31,42,45)(23,33,36,30,35,49,25,28)(24,37,41,52,40,29,46,48)(34,51,39,44,38,50,43,47);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,28,31,30)(24,38,40,34)(25,42,36,26)(29,47,37,44)(32,49,45,33)(39,46,43,41)(48,50,52,51);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,40)(22,29,31,37)(23,34,35,38)(25,43,36,39)(26,41,42,46)(28,44,30,47)(32,48,45,52)(33,50,49,51);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,36,42,35)(22,30,32,49,31,28,45,33)(24,39,41,38,40,43,46,34)(29,44,48,50,37,47,52,51);(6,12)(7,14)(11,20)(13,15)(22,31)(23,35)(24,40)(25,36)(32,45)(41,46)(44,47)(50,51)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=f,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, 6942219);