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 |
| prime radical | #46017 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4183 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)(9,10,15,17)(11,18,14,16)(12,13,22,24)(19,20,21,23)(25,26,32,36)(27,33,38,39)(28,29,30,34)(31,37,40,35);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,17,18)(12,21,22,19)(13,20,24,23)(25,27,32,38)(26,33,36,39)(28,35,30,37)(29,31,34,40);(9,12)(10,13)(11,19)(14,21)(15,22)(16,23)(17,24)(18,20)(25,28,32,30)(26,34,36,29)(27,37,38,35)(31,39,40,33);(9,13)(10,12)(11,20)(14,23)(15,24)(16,21)(17,22)(18,19)(25,29,32,34)(26,28,36,30)(27,31,38,40)(33,35,39,37);(1,4,5,2)(3,7,8,6)(9,12)(10,13)(11,21)(14,19)(15,22)(16,20)(17,24)(18,23)(25,30,32,28)(26,34,36,29)(27,35,38,37)(31,39,40,33);(9,14,15,11)(10,16,17,18)(12,21,22,19)(13,23,24,20)(25,31)(26,35)(27,34)(28,39)(29,38)(30,33)(32,40)(36,37)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=hi,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, 7965095);