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 | |
4344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10,15,16)(11,17,12,14)(13,21,23,18)(19,24,20,22)(25,26)(27,31)(28,36)(29,34)(30,32)(33,38)(35,40)(37,39);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,14,16,17)(13,19,23,20)(18,24,21,22)(25,27,30,29)(26,31,32,34)(28,35,38,37)(33,39,36,40);(10,16)(11,12)(18,21)(19,20)(26,32)(31,34)(33,36)(39,40);(1,2,5,4)(3,6,8,7)(9,12,15,11)(10,17,16,14)(13,20,23,19)(18,22,21,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,13,15,23)(10,18,16,21)(11,19,12,20)(14,24,17,22)(25,28)(26,33)(27,35)(29,37)(30,38)(31,39)(32,36)(34,40);(9,14,15,17)(10,11,16,12)(13,22,23,24)(18,20,21,19)(25,29,30,27)(26,34,32,31)(28,37,38,35)(33,40,36,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7861918);