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 | |
4424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,17,18,12)(13,21,22,16)(19,20,24,23)(25,26,31,36)(27,37,38,32)(28,33,35,30)(29,34,39,40);(1,2)(3,6)(4,7)(5,8)(10,15)(12,17)(16,21)(20,23)(27,38)(29,39)(32,37)(34,40);(1,3)(2,5)(4,8)(6,7)(10,15)(11,18)(13,22)(20,23)(25,27,31,38)(26,32,36,37)(28,39,35,29)(30,40,33,34);(3,8)(5,6)(9,11,14,18)(10,12,15,17)(13,19,22,24)(16,20,21,23)(25,28,31,35)(26,33,36,30)(27,29,38,39)(32,40,37,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,20)(14,17)(15,18)(16,19)(21,24)(22,23)(25,29)(26,34)(27,28)(30,32)(31,39)(33,37)(35,38)(36,40);(1,2)(3,6)(4,7)(5,8)(9,13)(10,16)(11,19)(12,20)(14,22)(15,21)(17,23)(18,24)(25,30,31,33)(26,35,36,28)(27,34,38,40)(29,32,39,37)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8811782);