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 | |
4537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,15)(11,19,20,16)(12,17,21,24)(13,22,23,18)(25,26)(27,30)(28,33)(29,34)(31,32)(35,40)(36,38)(37,39);(1,2)(3,6)(4,7)(5,8)(10,15)(12,21)(16,19)(18,22)(25,27)(26,30)(28,36)(29,37)(31,35)(32,40)(33,38)(34,39);(1,3)(2,5)(4,8)(6,7)(10,15)(11,20)(17,24)(18,22)(26,32)(27,35)(28,37)(34,38);(3,8)(5,6)(9,11)(10,16)(12,22)(13,17)(14,20)(15,19)(18,21)(23,24)(25,28,31,37)(26,33,32,39)(27,36,35,29)(30,38,40,34);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,18)(13,19)(14,21)(15,24)(16,23)(20,22)(25,29,31,36)(26,34,32,38)(27,28,35,37)(30,33,40,39);(1,2)(3,5)(4,7)(6,8)(9,13,14,23)(10,18,15,22)(11,17,20,24)(12,16,21,19)(25,30,31,40)(26,35,32,27)(28,38,37,34)(29,39,36,33)Polycyclic presentation:
2^9|a^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9033165);