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 | |
4552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,14)(15,17)(16,21)(22,23)(25,26)(27,34)(28,37)(29,40)(30,36)(31,35)(32,38)(33,39);(2,6)(3,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,31,38)(26,32,35,34)(28,36,39,29)(30,37,40,33);(1,3)(2,4)(5,7)(6,8)(12,20)(16,21)(18,24)(22,23)(25,28,31,39)(26,33,35,37)(27,36,38,29)(30,34,40,32);(1,4)(2,3)(5,8)(6,7)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,29)(26,30)(27,28)(31,36)(32,37)(33,34)(35,40)(38,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,34)(28,36)(29,39)(30,37)(31,38)(32,35)(33,40);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)(25,30)(26,29)(27,37)(28,32)(31,40)(33,38)(34,39)(35,36)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8800656);