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 | |
4279 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)(9,10,14,15)(11,16,19,22)(12,17,21,20)(13,23,24,18)(25,26,31,33)(27,29,34,32)(28,38,35,30)(36,37,40,39);(1,2,5,6)(3,7,4,8)(10,15)(12,21)(16,22)(18,23)(25,27)(26,32)(28,36)(29,33)(30,37)(31,34)(35,40)(38,39);(1,3,5,4)(2,7,6,8)(10,15)(11,19)(17,20)(18,23)(26,33)(27,34)(28,35)(37,39);(9,11,14,19)(10,16,15,22)(12,18,21,23)(13,20,24,17)(25,28,31,35)(26,30,33,38)(27,36,34,40)(29,39,32,37);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,17,15,20)(11,18,19,23)(13,22,24,16)(25,29,31,32)(26,34,33,27)(28,39,35,37)(30,40,38,36);(9,13)(10,18)(11,20)(12,22)(14,24)(15,23)(16,21)(17,19)(25,30)(26,35)(27,37)(28,33)(29,36)(31,38)(32,40)(34,39)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7944726);