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 | |
4575 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)(9,10,15,19)(11,16,20,23)(12,17,21,22)(13,14,18,24)(25,26,31,32)(27,30,33,40)(28,29,37,38)(34,39,36,35);(2,3)(6,7)(9,11,15,20)(10,16,19,23)(12,13,21,18)(14,22,24,17)(25,27)(26,30)(28,36)(29,35)(31,33)(32,40)(34,37)(38,39);(1,3,5,2)(4,7,8,6)(11,20)(12,21)(16,23)(17,22)(26,32)(27,33)(28,37)(35,39);(1,4)(2,6)(3,7)(5,8)(9,12,15,21)(10,17,19,22)(11,13,20,18)(14,23,24,16)(25,28)(26,29)(27,34)(30,39)(31,37)(32,38)(33,36)(35,40);(9,13)(10,14)(11,21)(12,20)(15,18)(16,22)(17,23)(19,24)(25,29)(26,28)(27,35)(30,36)(31,38)(32,37)(33,39)(34,40);(9,14)(10,18)(11,22)(12,16)(13,19)(15,24)(17,20)(21,23)(25,30,31,40)(26,33,32,27)(28,35,37,39)(29,34,38,36)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=ghi,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, 8625397);