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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,20)(12,22)(13,23)(14,19)(15,17)(16,21)(18,24)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,37)(38,39);(3,7)(5,8)(9,11,15,21)(10,16,17,20)(12,23,24,14)(13,22,19,18)(25,27)(26,32)(28,38)(29,39)(30,33)(31,35)(34,36)(37,40);(1,2)(3,5)(4,6)(7,8)(10,17)(13,19)(16,20)(18,22)(26,33)(27,35)(29,40)(36,38);(1,3,4,7)(2,5,6,8)(9,12,15,24)(10,18,17,22)(11,14,21,23)(13,16,19,20)(25,28,31,34)(26,29,33,40)(27,36,35,38)(30,39,32,37);(1,2)(3,5)(4,6)(7,8)(9,13,15,19)(10,14,17,23)(11,18,21,22)(12,16,24,20)(25,29)(26,34)(27,37)(28,33)(30,36)(31,40)(32,38)(35,39);(9,14)(10,19)(11,12)(13,17)(15,23)(16,18)(20,22)(21,24)(25,30)(26,35)(27,33)(28,37)(29,36)(31,32)(34,39)(38,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8494944);