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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,16)(12,17)(18,24)(20,22)(25,26)(27,30)(28,39)(29,31)(32,37)(33,36)(34,38)(35,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,22)(23,24)(25,27,32,36)(26,33,37,30)(28,31,38,35)(29,39,40,34);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,12,16,17)(14,19,23,21)(18,20,24,22)(25,28)(26,34)(27,31)(29,33)(30,40)(32,38)(35,36)(37,39);(1,3)(2,6)(4,8)(5,7)(9,12,15,17)(10,11,16,13)(14,22,23,20)(18,21,24,19)(25,29,32,40)(26,35,37,31)(27,34,36,39)(28,33,38,30);(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,22,24,20)(25,30,32,33)(26,36,37,27)(28,40,38,29)(31,34,35,39);(1,2)(3,5)(4,7)(6,8)(9,14,15,23)(10,18,16,24)(11,19,13,21)(12,20,17,22)(25,31,32,35)(26,29,37,40)(27,38,36,28)(30,34,33,39)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=g,e^2=g,f^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, 8328994);