Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1459 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)(4,8)(5,6)(17,18,22,26,23,27,21,34)(19,24,30,40,29,38,35,39)(20,31,32,42,33,41,36,25)(28,43,46,47,44,48,45,37);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,29)(18,24,27,38)(20,28,33,44)(21,35,22,30)(25,37,42,47)(26,40,34,39)(31,43,41,48)(32,46,36,45);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,33)(18,25,27,42)(19,28,29,44)(21,32,22,36)(24,37,38,47)(26,41,34,31)(30,45,35,46)(39,43,40,48);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(19,29)(24,38)(28,44)(30,35)(37,47)(39,40)(43,48)(45,46);(11,15)(14,16)(17,21,23,22)(18,26,27,34)(19,30,29,35)(20,32,33,36)(24,39,38,40)(25,41,42,31)(28,45,44,46)(37,43,47,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6456994);