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 | |
4635 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,4)(6,7)(9,10)(11,16)(12,20)(13,17)(14,18)(15,21)(19,23)(22,24)(25,26,31,35)(27,32,36,39)(28,33,38,37)(29,30,34,40);(1,2,5,8)(3,6,4,7)(9,11)(10,13)(12,19)(14,17)(15,23)(16,18)(20,24)(21,22)(25,27)(26,32)(28,30)(29,37)(31,36)(33,34)(35,39)(38,40);(1,3,5,4)(2,6,8,7)(12,21)(15,20)(19,22)(23,24)(27,36)(28,38)(32,39)(33,37);(9,12,14,21)(10,15,18,20)(11,19,17,22)(13,23,16,24)(25,28)(26,33)(27,30)(29,39)(31,38)(32,34)(35,37)(36,40);(1,2)(3,6)(4,7)(5,8)(9,11)(10,16)(12,22)(13,18)(14,17)(15,23)(19,21)(20,24)(25,29)(26,30)(27,37)(28,32)(31,34)(33,36)(35,40)(38,39);(1,4,5,3)(2,7,8,6)(9,13,14,16)(10,17,18,11)(12,23,21,24)(15,22,20,19)(25,30)(26,34)(27,28)(29,35)(31,40)(32,33)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9035840);