Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1485 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(17,18,22,27)(19,28,31,24)(20,32,34,46)(21,26,23,38)(25,35,43,36)(29,44,33,47)(30,41,37,40)(39,45,42,48);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,22,31)(18,24,27,28)(20,33,34,29)(21,37,23,30)(25,42,43,39)(26,41,38,40)(32,44,46,47)(35,45,36,48);(9,11,13,12)(10,15,14,16)(17,20,23,36,22,34,21,35)(18,25,26,32,27,43,38,46)(19,29,30,45,31,33,37,48)(24,39,41,47,28,42,40,44);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(20,34)(25,43)(29,33)(32,46)(35,36)(39,42)(44,47)(45,48);(9,12,13,11)(10,16,14,15)(17,21,22,23)(18,26,27,38)(19,30,31,37)(20,35,34,36)(24,40,28,41)(25,32,43,46)(29,45,33,48)(39,44,42,47)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=ghi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6535866);