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 | |
4291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,12,18,16)(13,17,22,23)(19,20,21,24)(25,26,31,32)(27,28,35,40)(29,33,30,34)(36,38,37,39);(1,2)(3,5)(4,6)(7,8)(10,15)(11,18)(17,23)(19,21)(26,32)(27,35)(33,34)(36,37);(1,2,4,6)(3,5,7,8)(10,15)(12,16)(17,23)(20,24)(25,27)(26,28)(29,36)(30,37)(31,35)(32,40)(33,38)(34,39);(3,7)(5,8)(9,11)(10,16)(12,15)(13,21)(14,18)(17,20)(19,22)(23,24)(25,28)(26,27)(29,38)(30,39)(31,40)(32,35)(33,36)(34,37);(1,2)(3,5)(4,6)(7,8)(9,12,14,16)(10,11,15,18)(13,20,22,24)(17,19,23,21)(25,29,31,30)(26,33,32,34)(27,36,35,37)(28,38,40,39);(1,3,4,7)(2,5,6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,30,31,29)(26,34,32,33)(27,37,35,36)(28,39,40,38)Polycyclic presentation:
2^9|a^2=g,c^2=ghi,e^2=g,f^2=i,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, 8459548);