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 | |
4636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(3,7)(5,8)(9,10,15,17)(11,19,21,16)(12,20,23,18)(13,22,24,14)(25,26,31,34)(27,32,35,29)(28,30,33,39)(36,37,40,38);(1,2)(3,8)(4,6)(5,7)(9,11,15,21)(10,16,17,19)(12,22,23,14)(13,18,24,20)(25,27)(26,32)(28,36)(29,34)(30,37)(31,35)(33,40)(38,39);(1,3,4,7)(2,5,6,8)(10,17)(13,24)(16,19)(18,20)(27,35)(29,32)(36,40)(37,38);(1,2,4,6)(3,5,7,8)(9,12,15,23)(10,18,17,20)(11,14,21,22)(13,19,24,16)(25,28)(26,30)(27,36)(29,38)(31,33)(32,37)(34,39)(35,40);(1,3)(2,5)(4,7)(6,8)(9,13)(10,14)(11,20)(12,19)(15,24)(16,23)(17,22)(18,21)(25,29)(26,27)(28,38)(30,36)(31,32)(33,37)(34,35)(39,40);(1,2)(3,5)(4,6)(7,8)(9,14)(10,13)(11,12)(15,22)(16,18)(17,24)(19,20)(21,23)(25,30)(26,33)(27,37)(28,34)(29,36)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8495179);