Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | #37700 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1514 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)(3,6)(4,8)(5,7)(17,18)(19,28)(20,33)(21,27)(22,24)(23,38)(25,32)(26,36)(29,46)(30,39)(31,44)(34,45)(35,43)(37,42)(40,48)(41,47);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(18,24)(25,39)(26,33)(27,38)(28,44)(40,41)(42,43)(45,46);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,32,22,31,21,30)(18,25,27,28,24,39,38,44)(20,34,35,48,36,29,37,47)(26,41,43,46,33,40,42,45);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,20,22,36)(18,26,24,33)(19,29,31,34)(21,37,23,35)(25,40,39,41)(27,43,38,42)(28,45,44,46)(30,48,32,47);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,24,38)(19,30,31,32)(20,35,36,37)(25,28,39,44)(26,42,33,43)(29,47,34,48)(40,46,41,45)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6821805);