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 | |
4369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(10,15)(12,19)(17,21)(20,24)(25,26,31,35)(27,32,37,29)(28,36,39,33)(30,38,40,34);(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,27,31,37)(26,32,35,29)(28,38,39,34)(30,36,40,33);(1,3,6,5)(2,7,4,8)(9,11,14,16)(10,12,15,19)(13,18,22,23)(17,20,21,24)(25,28,31,39)(26,33,35,36)(27,34,37,38)(29,30,32,40);(1,3,6,5)(2,8,4,7)(9,10)(11,12)(13,21)(14,15)(16,19)(17,22)(18,24)(20,23)(25,29)(26,27)(28,30)(31,32)(33,34)(35,37)(36,38)(39,40);(1,4,6,2)(3,8,5,7)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,21)(22,24)(25,26)(27,29)(28,33)(30,34)(31,35)(32,37)(36,39)(38,40);(1,5,6,3)(2,8,4,7)(9,13,14,22)(10,17,15,21)(11,18,16,23)(12,20,19,24)(25,30)(26,34)(27,36)(28,32)(29,39)(31,40)(33,37)(35,38)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10358855);