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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,14)(12,20)(13,16)(15,21)(17,19)(18,23)(22,24)(25,26,31,32)(27,37,39,33)(28,36,30,34)(29,35,40,38);(1,2,5,6)(3,4,7,8)(9,10)(11,17)(12,20)(13,16)(14,19)(15,21)(18,22)(23,24)(26,32)(28,30)(33,37)(35,38);(2,6)(3,7)(9,10,13,16)(11,14,19,17)(12,15,21,20)(18,22,24,23)(25,27,31,39)(26,33,32,37)(28,35,30,38)(29,34,40,36);(2,6)(3,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,28,31,30)(26,34,32,36)(27,35,39,38)(29,33,40,37);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,24)(25,29)(26,35)(27,36)(28,33)(30,37)(31,40)(32,38)(34,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(25,30,31,28)(26,36,32,34)(27,38,39,35)(29,37,40,33)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10235862);