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 | |
4368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10)(11,19)(12,22)(13,18)(14,16)(15,21)(17,23)(20,24)(25,26,31,32)(27,28,33,36)(29,37,38,30)(34,40,39,35);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,16,19)(12,20,23,18)(13,17,24,22)(25,27,31,33)(26,28,32,36)(29,34,38,39)(30,35,37,40);(2,5)(6,8)(10,16)(12,23)(15,19)(18,20)(26,32)(27,33)(29,38)(35,40);(3,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(25,28,31,36)(26,33,32,27)(29,35,38,40)(30,39,37,34);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,19)(14,23)(15,24)(16,22)(18,21)(25,29)(26,30)(27,34)(28,35)(31,38)(32,37)(33,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,24)(10,18,16,20)(11,17,21,22)(12,19,23,15)(25,30)(26,29)(27,35)(28,34)(31,37)(32,38)(33,40)(36,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=gh,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, 8416958);