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 | |
4608 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,17)(13,18)(15,19)(16,21)(20,22)(23,24)(26,30)(28,35)(31,37)(33,36);(1,2,4,6)(3,5,8,7)(10,11)(15,17)(16,18)(22,23)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40);(1,2)(3,5)(4,6)(7,8)(10,11)(12,19)(16,18)(20,24)(25,27,29,34)(26,31,30,37)(28,36,35,33)(32,40,38,39);(1,3,4,8)(2,5,6,7)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,28,29,35)(26,32,30,38)(27,33,34,36)(31,39,37,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,27)(26,31)(28,33)(29,34)(30,37)(32,39)(35,36)(38,40);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,26)(27,31)(28,32)(29,30)(33,39)(34,37)(35,38)(36,40)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8034410);