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 | |
4713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,21)(16,19)(22,23)(25,26)(27,34)(28,38)(29,33)(30,35)(31,32)(36,37)(39,40);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(26,32)(28,35)(33,34)(36,39);(1,3)(2,5)(4,8)(6,7)(25,27,31,29)(26,33,32,34)(28,36,35,39)(30,37,38,40);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,28)(26,30)(27,36)(29,39)(31,35)(32,38)(33,37)(34,40);(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,29,31,27)(26,34,32,33)(28,39,35,36)(30,40,38,37);(1,2)(3,5)(4,7)(6,8)(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30)(26,35)(27,37)(28,32)(29,40)(31,38)(33,39)(34,36)Polycyclic presentation:
2^9|c^2=h,e^2=g,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, 8328886);