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 | |
4423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,16)(11,13,12,17)(14,18,22,23)(19,21,20,24)(25,26,31,35)(27,29,36,32)(28,33,37,38)(30,34,39,40);(1,2)(3,5)(4,6)(7,8)(10,16)(11,12)(18,23)(19,20)(27,36)(28,37)(29,32)(33,38);(2,3)(6,7)(9,11,15,12)(10,17,16,13)(14,19,22,20)(18,24,23,21)(25,27,31,36)(26,32,35,29)(28,30,37,39)(33,40,38,34);(4,8)(6,7)(9,12,15,11)(10,13,16,17)(14,20,22,19)(18,21,23,24)(25,28)(26,33)(27,30)(29,34)(31,37)(32,40)(35,38)(36,39);(1,3,5,2)(4,7,8,6)(9,13)(10,11)(12,16)(14,21)(15,17)(18,19)(20,23)(22,24)(25,29,31,32)(26,27,35,36)(28,34,37,40)(30,38,39,33);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,18,16,23)(11,19,12,20)(13,21,17,24)(25,30)(26,34)(27,37)(28,36)(29,38)(31,39)(32,33)(35,40)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=g,e^2=gi,f^2=g,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, 8519373);