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 | |
4714 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,20)(13,17)(14,19)(15,21)(18,22)(23,24)(25,26,29,32)(27,33,35,30)(28,31,38,37)(34,40,36,39);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,29,35)(26,30,32,33)(28,34,38,36)(31,39,37,40);(1,2)(3,5)(4,6)(7,8)(12,21)(15,20)(18,24)(22,23)(25,27,29,35)(26,30,32,33)(28,34,38,36)(31,39,37,40);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,23)(17,20)(19,24)(25,27)(26,30)(28,36)(29,35)(31,40)(32,33)(34,38)(37,39);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,22)(21,24)(25,26)(27,30)(28,37)(29,32)(31,38)(33,35)(34,40)(36,39);(9,10)(11,14)(12,20)(13,17)(15,21)(16,19)(18,23)(22,24)(25,28)(26,31)(27,34)(29,38)(30,39)(32,37)(33,40)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616466);