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 | |
4362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,16,20)(11,21,23,17)(12,15,13,18)(14,19,24,22)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,2)(3,7)(4,5)(6,8)(9,11,16,23)(10,17,20,21)(12,22,13,19)(14,15,24,18)(25,27,31,37)(26,32,35,30)(28,36,39,29)(33,34,40,38);(2,5)(6,7)(9,12,16,13)(10,18,20,15)(11,22,23,19)(14,21,24,17)(25,28,31,39)(26,33,35,40)(27,36,37,29)(30,38,32,34);(2,5)(6,7)(9,13,16,12)(10,15,20,18)(11,22,23,19)(14,21,24,17)(25,27)(26,32)(28,36)(29,39)(30,35)(31,37)(33,34)(38,40);(1,2)(3,6)(4,5)(7,8)(9,14)(10,19)(11,15)(12,17)(13,21)(16,24)(18,23)(20,22)(25,29,31,36)(26,34,35,38)(27,28,37,39)(30,33,32,40);(1,3)(2,6)(4,8)(5,7)(9,15)(10,12)(11,14)(13,20)(16,18)(17,19)(21,22)(23,24)(25,30,31,32)(26,27,35,37)(28,38,39,34)(29,33,36,40)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10231283);