Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,34)(28,42)(29,38)(30,40)(32,45)(33,41)(35,39)(37,44)(43,48)(46,47);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,30)(18,24,27,40)(20,28,32,43)(21,35,22,29)(25,37,41,47)(26,39,36,38)(31,42,45,48)(33,46,34,44);(10,13)(14,16)(17,20,21,33,23,32,22,34)(18,25,36,45,27,41,26,31)(19,28,35,46,30,43,29,44)(24,37,38,48,40,47,39,42);(1,3,5,4)(2,7,8,6)(9,11)(10,14)(12,15)(13,16)(20,32)(25,41)(28,43)(31,45)(33,34)(37,47)(42,48)(44,46);(1,4,5,3)(2,6,8,7)(11,15)(14,16)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,33,32,34)(24,38,40,39)(25,31,41,45)(28,44,43,46)(37,48,47,42)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6466926);