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 | |
4416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,21)(16,19)(22,23)(26,31)(27,36)(28,35)(38,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,26)(27,32)(28,34)(29,35)(30,31)(33,36)(37,39)(38,40);(1,3,5,6)(2,4,7,8)(25,27)(26,32)(28,37)(29,38)(30,36)(31,33)(34,39)(35,40);(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,27)(26,33)(28,37)(29,39)(30,36)(31,32)(34,38)(35,40);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,28,30,35)(26,34,31,29)(27,37,36,40)(32,39,33,38);(1,4)(2,6)(3,7)(5,8)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,29,30,34)(26,35,31,28)(27,38,36,39)(32,40,33,37)Polycyclic presentation:
2^9|c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8331880);