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 | |
4520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,5)(3,4,8,7)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26,31,34)(27,29,32,37)(28,30,33,39)(35,36,38,40);(1,3)(2,7)(4,5)(6,8)(9,11)(10,16)(12,22)(13,23)(14,17)(15,19)(18,20)(21,24)(25,27,31,32)(26,29,34,37)(28,35,33,38)(30,36,39,40);(1,4,6,7)(2,3,5,8)(10,17)(11,19)(13,24)(20,22)(27,32)(29,37)(35,38)(36,40);(2,5)(3,8)(9,12,15,18)(10,13,17,24)(11,20,19,22)(14,23,16,21)(25,28)(26,30)(27,35)(29,36)(31,33)(32,38)(34,39)(37,40);(1,4)(2,3)(5,8)(6,7)(9,13)(10,18)(11,21)(12,17)(14,20)(15,24)(16,22)(19,23)(25,29,31,37)(26,32,34,27)(28,36,33,40)(30,38,39,35);(1,5,6,2)(3,4,8,7)(9,14)(10,19)(11,17)(12,21)(13,20)(15,16)(18,23)(22,24)(25,30)(26,33)(27,36)(28,34)(29,38)(31,39)(32,40)(35,37)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8528920);