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 | |
4268 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)(6,8)(10,14)(11,17)(16,21)(18,23)(25,26)(27,35)(28,33)(29,32)(30,34)(31,39)(36,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27)(26,30)(28,36)(29,35)(31,37)(32,34)(33,40)(38,39);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,28,32,31)(26,33,29,39)(27,36,34,37)(30,40,35,38);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,29,32,26)(27,35,34,30)(28,33,31,39)(36,40,37,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,30,32,35)(26,34,29,27)(28,38,31,40)(33,36,39,37);(1,3,4,7)(2,6,5,8)(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,31,32,28)(26,33,29,39)(27,37,34,36)(30,40,35,38)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=gh,e^2=g,f^2=ghi,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^c=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8040017);