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 | |
4430 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(9,10)(11,16)(12,22)(13,17)(14,18)(15,19)(20,24)(21,23)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(3,7)(6,8)(9,11,15,20)(10,16,19,24)(12,23,17,14)(13,18,22,21)(26,30)(27,34)(32,36)(35,40);(1,2)(3,6)(4,5)(7,8)(11,20)(14,23)(16,24)(18,21)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(9,12)(10,13)(11,14)(15,17)(16,21)(18,24)(19,22)(20,23)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,2)(3,6)(4,5)(7,8)(9,13)(10,17)(11,21)(12,19)(14,24)(15,22)(16,23)(18,20)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39);(9,14)(10,18)(11,12)(13,24)(15,23)(16,22)(17,20)(19,21)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313892);