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 | |
4537 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)(3,4)(5,6)(7,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26)(27,32)(28,29)(30,35)(31,33)(34,40)(36,39)(37,38);(2,6)(3,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,31,35)(26,32,33,30)(28,38,34,36)(29,37,40,39);(1,3,5,7)(2,4,6,8)(12,21)(15,20)(18,24)(22,23)(26,33)(27,35)(28,34)(37,39);(1,4)(2,7)(3,6)(5,8)(9,12,13,21)(10,15,17,20)(11,18,19,24)(14,22,16,23)(25,28,31,34)(26,29,33,40)(27,36,35,38)(30,37,32,39);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,29)(26,34)(27,37)(28,33)(30,38)(31,40)(32,36)(35,39);(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,30,31,32)(26,35,33,27)(28,39,34,37)(29,36,40,38)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^2=gi,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, 8626156);