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 | |
4697 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(3,7)(9,10,14,15)(11,19,20,16)(12,22,23,17)(13,18,24,21)(25,26)(27,36)(28,30)(29,39)(31,34)(32,35)(33,38)(37,40);(1,2,4,5)(3,6,7,8)(10,15)(11,20)(17,22)(18,21)(25,27)(26,33)(28,37)(29,34)(30,31)(32,38)(35,36)(39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(12,23)(16,19)(18,21)(25,28,32,29)(26,30,35,39)(27,37,38,34)(31,36,40,33);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,23)(22,24)(25,29,32,28)(26,30,35,39)(27,34,38,37)(31,36,40,33);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21)(13,16)(14,23)(15,22)(18,20)(19,24)(25,30)(26,28)(27,31)(29,35)(32,39)(33,37)(34,36)(38,40);(9,13)(10,18)(11,22)(12,16)(14,24)(15,21)(17,20)(19,23)(25,31)(26,34)(27,30)(28,36)(29,33)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=gh,d^2=gh,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, 8429345);