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 | |
4494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,6)(7,8)(10,15)(11,18)(17,21)(19,24)(26,30)(27,33)(32,36)(35,37);(1,2)(3,7)(4,8)(5,6)(9,10,14,15)(11,16,18,12)(13,21,23,17)(19,20,24,22)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(10,15)(12,16)(17,21)(20,22)(25,26,29,30)(27,34,33,31)(28,32,38,36)(35,40,37,39);(9,11)(10,16)(12,15)(13,19)(14,18)(17,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,7,6,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(1,4,5,3)(2,8,6,7)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,28,29,38)(26,32,30,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|b^2=hi,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7876874);