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 | |
4440 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)(12,20)(13,15)(14,19)(16,22)(18,23)(21,24)(25,26,31,32)(27,29,35,38)(28,33,37,39)(30,34,36,40);(1,2)(3,4)(5,6)(7,8)(9,11,13,19)(10,14,15,17)(12,21,22,18)(16,24,20,23)(25,27,31,35)(26,29,32,38)(28,36,37,30)(33,40,39,34);(2,6)(4,7)(10,15)(14,17)(16,20)(23,24)(26,32)(27,35)(30,36)(33,39);(9,12,13,22)(10,16,15,20)(11,18,19,21)(14,23,17,24)(25,28)(26,33)(27,30)(29,34)(31,37)(32,39)(35,36)(38,40);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,24)(21,22)(25,29)(26,27)(28,34)(30,33)(31,38)(32,35)(36,39)(37,40);(1,4,5,7)(2,3,6,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,30)(26,34)(27,28)(29,33)(31,36)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=g,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, 8139674);