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 | |
4426 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)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,31,33)(27,32,28,36)(29,30,39,40)(34,38,37,35);(2,6)(3,7)(10,14)(11,17)(16,21)(19,23)(25,27,31,28)(26,32,33,36)(29,37,39,34)(30,35,40,38);(1,3)(2,4)(5,7)(6,8)(12,20)(16,21)(19,23)(22,24)(25,28,31,27)(26,32,33,36)(29,37,39,34)(30,38,40,35);(1,4)(2,3)(5,8)(6,7)(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,29)(26,30)(27,34)(28,37)(31,39)(32,38)(33,40)(35,36);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,30)(26,29)(27,35)(28,38)(31,40)(32,37)(33,39)(34,36);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,f^2=gi,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, 8325640);