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 | |
4424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,20)(18,22)(19,23)(21,24)(25,26)(27,29)(28,36)(30,31)(32,38)(33,34)(35,40)(37,39);(2,3)(6,7)(9,11,15,13)(10,16,12,17)(14,19,22,21)(18,23,20,24)(25,27)(26,29)(28,35)(30,33)(31,34)(32,37)(36,40)(38,39);(1,2,5,3)(4,6,8,7)(10,12)(16,17)(18,20)(23,24)(26,31)(27,33)(32,36)(35,39);(1,3,5,2)(4,7,8,6)(9,12,15,10)(11,16,13,17)(14,20,22,18)(19,23,21,24)(25,26)(27,34)(28,36)(29,33)(30,31)(32,38)(35,37)(39,40);(1,4,5,8)(2,6,3,7)(9,13,15,11)(10,17,12,16)(14,21,22,19)(18,24,20,23)(25,28,30,38)(26,32,31,36)(27,35,33,39)(29,37,34,40);(9,14)(10,18)(11,19)(12,20)(13,21)(15,22)(16,23)(17,24)(25,29,30,34)(26,33,31,27)(28,37,38,40)(32,39,36,35)Polycyclic presentation:
2^9|b^2=hi,c^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066435);