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 | |
4678 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)(3,5)(4,8)(6,7)(9,10,14,15)(11,17,19,12)(13,16,21,22)(18,23,24,20)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(2,7)(3,4)(10,15)(12,17)(16,22)(20,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(2,7)(5,8)(13,21)(16,22)(18,24)(20,23)(26,30)(31,33)(32,36)(38,39);(1,3,6,4)(2,8,7,5)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39);(1,4,6,3)(2,5,7,8)(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,21)(10,16,15,22)(11,18,19,24)(12,20,17,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40)Polycyclic presentation:
2^9|a^2=ghi,d^2=h,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8161074);