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 | |
4645 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)(5,8)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,31,32)(27,29,35,38)(28,33,37,39)(30,34,36,40);(1,2,4,6)(3,5,7,8)(10,12)(15,17)(16,20)(22,23)(25,27,31,35)(26,29,32,38)(28,36,37,30)(33,40,39,34);(1,3)(2,5)(4,7)(6,8)(13,21)(16,20)(18,24)(22,23)(26,32)(27,35)(30,36)(33,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28)(26,33)(27,30)(29,34)(31,37)(32,39)(35,36)(38,40);(1,3)(2,5)(4,7)(6,8)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29)(26,27)(28,34)(30,33)(31,38)(32,35)(36,39)(37,40);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30)(26,34)(27,28)(29,33)(31,36)(32,40)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8108906);