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 | #52656 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,15,12)(11,16,13,17)(14,18,22,20)(19,23,21,24)(25,26)(27,36)(28,29)(30,34)(31,35)(32,37)(33,39)(38,40);(2,7)(3,4)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27,31,37)(26,32,35,36)(28,38,33,30)(29,34,39,40);(1,3,6,4)(2,8,7,5)(9,11,15,13)(10,17,12,16)(14,19,22,21)(18,24,20,23)(25,27,31,37)(26,32,35,36)(28,38,33,30)(29,34,39,40);(1,4,6,3)(2,5,7,8)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,28,31,33)(26,29,35,39)(27,30,37,38)(32,40,36,34);(1,5)(2,3)(4,7)(6,8)(9,13,15,11)(10,16,12,17)(14,21,22,19)(18,23,20,24)(25,29)(26,33)(27,34)(28,35)(30,32)(31,39)(36,38)(37,40);(9,14,15,22)(10,18,12,20)(11,19,13,21)(16,23,17,24)(25,30)(26,34)(27,28)(29,36)(31,38)(32,39)(33,37)(35,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8646635);