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 | |
4571 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19)(11,13,18,15)(12,16,22,17)(20,21,24,23)(25,26,31,35)(27,28,38,32)(29,39,37,33)(30,34,36,40);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,20)(13,19)(14,18)(16,23)(17,21)(22,24)(25,27)(26,32)(28,35)(29,36)(30,37)(31,38)(33,40)(34,39);(1,2)(3,5)(4,6)(7,8)(9,12,14,22)(10,16,19,17)(11,20,18,24)(13,21,15,23)(25,28,31,32)(26,27,35,38)(29,34,37,40)(30,33,36,39);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,19)(18,24)(25,28)(26,27)(29,34)(30,39)(31,32)(33,36)(35,38)(37,40);(9,11)(10,13)(12,20)(14,18)(15,19)(16,21)(17,23)(22,24)(25,29)(26,33)(27,36)(28,34)(30,38)(31,37)(32,40)(35,39);(9,13)(10,18)(11,19)(12,21)(14,15)(16,24)(17,20)(22,23)(25,30,31,36)(26,34,35,40)(27,37,38,29)(28,33,32,39)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8627799);