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 | |
4414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,21)(16,19)(22,24)(25,26,31,32)(27,28,33,36)(29,37,38,30)(34,40,39,35);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,31,33)(26,28,32,36)(29,34,38,39)(30,35,37,40);(2,5)(6,8)(11,17)(15,21)(18,23)(22,24)(26,32)(27,33)(29,38)(35,40);(3,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,28,31,36)(26,33,32,27)(29,35,38,40)(30,39,37,34);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29)(26,30)(27,34)(28,35)(31,38)(32,37)(33,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30)(26,29)(27,35)(28,34)(31,37)(32,38)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8416965);