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 | |
4377 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,16)(13,17)(15,18)(19,22)(20,21)(23,24)(25,26,31,28)(27,35,37,32)(29,38,36,33)(30,34,40,39);(1,3,6,5)(2,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,31,37)(26,32,28,35)(29,30,36,40)(33,34,38,39);(1,3,6,5)(2,8,4,7)(11,18)(14,15)(19,24)(22,23)(26,28)(32,35)(33,38)(34,39);(1,4,6,2)(3,7,5,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24)(25,28,31,26)(27,32,37,35)(29,38,36,33)(30,34,40,39);(1,5,6,3)(2,7,4,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29)(26,33)(27,30)(28,38)(31,36)(32,34)(35,39)(37,40);(9,10)(11,14)(12,20)(13,17)(15,18)(16,21)(19,23)(22,24)(25,30)(26,34)(27,36)(28,39)(29,37)(31,40)(32,38)(33,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8635789);