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 | |
4346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,3)(2,6)(4,7)(5,8)(11,17)(15,21)(18,23)(22,24)(26,30)(27,34)(32,36)(35,40);(1,3,5,8)(2,6,7,4)(10,14)(15,21)(16,19)(22,24)(27,34)(31,33)(35,40)(38,39);(3,8)(4,6)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,2)(3,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10224039);