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 | |
4433 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10,14,15)(11,19,20,16)(12,17,22,24)(13,21,23,18)(25,26,31,34)(27,35,38,32)(28,39,36,30)(29,33,40,37);(1,3,5,8)(2,6,7,4)(10,15)(11,20)(17,24)(18,21)(25,27,31,38)(26,32,34,35)(28,40,36,29)(30,37,39,33);(1,3)(2,4)(5,8)(6,7)(10,15)(11,20)(12,22)(13,23)(28,36)(29,40)(30,39)(33,37);(1,4)(2,3)(5,6)(7,8)(9,11,14,20)(10,16,15,19)(12,18,22,21)(13,17,23,24)(25,28)(26,30)(27,29)(31,36)(32,33)(34,39)(35,37)(38,40);(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,19,23,16)(25,29)(26,33)(27,36)(28,38)(30,35)(31,40)(32,39)(34,37);(9,13)(10,18)(11,17)(12,19)(14,23)(15,21)(16,22)(20,24)(25,30)(26,28)(27,37)(29,35)(31,39)(32,40)(33,38)(34,36)Polycyclic presentation:
2^9|a^2=g,b^2=gh,d^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,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, 8966367);