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 | |
4695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,39)(37,40);(1,2)(3,7)(4,5)(6,8)(10,14)(15,17)(16,21)(22,23)(25,27,29,35)(26,30,31,33)(28,37,38,34)(32,40,36,39);(2,5)(6,7)(12,20)(16,21)(18,24)(22,23)(26,31)(30,33)(32,36)(39,40);(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28,29,38)(26,32,31,36)(27,34,35,37)(30,39,33,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,26)(27,30)(28,32)(29,31)(33,35)(34,39)(36,38)(37,40)Polycyclic presentation:
2^9|b^2=h,d^2=g,f^2=gh,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, 8106841);