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 | |
4485 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,5)(6,8)(10,15)(11,19)(17,21)(18,20)(26,30)(27,33)(32,36)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,16,19,23)(12,17,22,21)(13,20,24,18)(25,26)(27,31)(28,36)(29,30)(32,38)(33,34)(35,40)(37,39);(10,15)(12,22)(16,23)(18,20)(25,26,29,30)(27,34,33,31)(28,32,38,36)(35,40,37,39);(9,11)(10,16)(12,20)(13,21)(14,19)(15,23)(17,24)(18,22)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(9,12)(10,17)(11,20)(13,23)(14,22)(15,21)(16,24)(18,19)(25,27,29,33)(26,31,30,34)(28,37,38,35)(32,40,36,39);(1,3,4,7)(2,6,5,8)(9,13,14,24)(10,18,15,20)(11,21,19,17)(12,16,22,23)(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=h,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, 7877461);