Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #47955 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4353 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,16,15)(11,21,23,17)(12,18,13,19)(14,20,22,24)(25,26)(27,30)(28,29)(31,39)(32,34)(33,40)(35,36)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11,16,23)(10,17,15,21)(12,22,13,14)(18,20,19,24)(25,27,32,29)(26,28,34,30)(31,35,40,37)(33,36,39,38);(1,2)(3,4)(5,8)(6,7)(9,12,16,13)(10,18,15,19)(11,22,23,14)(17,20,21,24)(25,28,32,30)(26,27,34,29)(31,36,40,38)(33,35,39,37);(1,4)(2,7)(3,8)(5,6)(9,13,16,12)(10,19,15,18)(11,14,23,22)(17,24,21,20)(25,29,32,27)(26,28,34,30)(31,37,40,35)(33,36,39,38);(1,4)(2,3)(5,6)(7,8)(9,14)(10,20)(11,12)(13,23)(15,24)(16,22)(17,19)(18,21)(25,30,32,28)(26,27,34,29)(31,36,40,38)(33,37,39,35);(1,2)(3,6)(4,7)(5,8)(9,15,16,10)(11,21,23,17)(12,19,13,18)(14,20,22,24)(25,31)(26,33)(27,35)(28,36)(29,37)(30,38)(32,40)(34,39)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9623439);