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 | |
4587 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,18)(11,12,16,15)(13,17,22,21)(19,20,24,23)(25,26,31,36)(27,37,39,32)(28,34,29,33)(30,35,38,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,16)(10,15,18,12)(13,19,22,24)(17,23,21,20)(25,27,31,39)(26,32,36,37)(28,38,29,30)(33,40,34,35);(1,3)(2,5)(4,8)(6,7)(25,28,31,29)(26,33,36,34)(27,38,39,30)(32,40,37,35);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,20)(23,24)(25,26)(27,37)(28,33)(29,34)(30,40)(31,36)(32,39)(35,38);(9,12,14,15)(10,16,18,11)(13,20,22,23)(17,24,21,19)(25,29,31,28)(26,34,36,33)(27,30,39,38)(32,35,37,40);(1,2)(3,5)(4,7)(6,8)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,30)(26,35)(27,28)(29,39)(31,38)(32,33)(34,37)(36,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8328889);