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 | #35782 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,7,5)(25,26,31,34)(27,28,33,32)(29,37,40,30)(35,39,38,36);(1,3)(2,5)(4,8)(6,7)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27,31,33)(26,32,34,28)(29,38,40,35)(30,36,37,39);(2,4)(5,8)(10,14)(12,21)(15,17)(18,22)(25,28,31,32)(26,33,34,27)(29,36,40,39)(30,38,37,35);(2,4)(3,7)(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,29)(26,30)(27,35)(28,36)(31,40)(32,39)(33,38)(34,37);(1,4,6,2)(3,8,7,5)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,28)(26,33)(27,34)(29,39)(30,35)(31,32)(36,40)(37,38);(1,5,6,8)(2,7,4,3)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,30)(26,29)(27,36)(28,35)(31,37)(32,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948143);