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 | #35539 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4354 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)(9,10,15,17)(11,21,22,16)(12,14,20,18)(13,24,23,19)(25,26)(27,33)(28,34)(29,39)(30,36)(31,37)(32,38)(35,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,23)(13,20)(14,24)(15,22)(17,21)(18,19)(25,27)(26,33)(28,38)(29,36)(30,39)(31,40)(32,34)(35,37);(10,17)(11,22)(12,20)(19,24)(25,28,32,33)(26,34,38,27)(29,30,40,37)(31,39,36,35);(1,3,4,7)(2,6,5,8)(9,12,15,20)(10,18,17,14)(11,23,22,13)(16,19,21,24)(25,29,32,40)(26,35,38,39)(27,36,34,31)(28,30,33,37);(9,13,15,23)(10,19,17,24)(11,20,22,12)(14,16,18,21)(25,30)(26,36)(27,39)(28,40)(29,33)(31,38)(32,37)(34,35);(9,14,15,18)(10,20,17,12)(11,24,22,19)(13,21,23,16)(25,31,32,36)(26,37,38,30)(27,40,34,29)(28,39,33,35)Polycyclic presentation:
2^9|a^2=h,c^2=hi,d^2=ghi,e^2=h,f^2=i,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, 7813184);