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 | |
4573 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10)(11,12)(13,17)(14,15)(16,18)(19,20)(21,23)(22,24)(25,26,31,35)(27,32,36,39)(28,33,37,38)(29,30,34,40);(2,3)(6,7)(9,11)(10,12)(13,19)(14,16)(15,18)(17,20)(21,22)(23,24)(25,27,31,36)(26,32,35,39)(28,29,37,34)(30,38,40,33);(1,3,5,2)(4,7,8,6)(10,15)(11,16)(17,21)(19,24)(27,36)(28,37)(32,39)(33,38);(1,4)(2,6)(3,7)(5,8)(9,10)(11,18)(12,16)(13,21)(14,15)(17,23)(19,20)(22,24)(25,28,31,37)(26,33,35,38)(27,29,36,34)(30,39,40,32);(9,12,14,18)(10,16,15,11)(13,22,23,20)(17,19,21,24)(25,29)(26,30)(27,37)(28,36)(31,34)(32,38)(33,39)(35,40);(9,13,14,23)(10,17,15,21)(11,19,16,24)(12,20,18,22)(25,30)(26,34)(27,38)(28,32)(29,35)(31,40)(33,36)(37,39)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=g,d^2=ghi,e^2=h,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, 8625634);