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 | |
4431 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)(11,12)(13,21)(14,15)(16,19)(17,22)(18,23)(20,24)(25,26,31,33)(27,34,37,30)(28,32,38,29)(35,36,40,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,16)(17,21)(20,23)(25,27)(26,30)(28,35)(29,36)(31,37)(32,39)(33,34)(38,40);(2,5)(3,7)(9,10,14,15)(11,16,19,12)(13,17,22,21)(18,23,24,20)(25,28,31,38)(26,32,33,29)(27,35,37,40)(30,39,34,36);(3,7)(6,8)(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,21)(22,24)(25,26)(27,30)(28,32)(29,38)(31,33)(34,37)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,11,15,19)(13,20,22,23)(17,18,21,24)(25,29,31,32)(26,28,33,38)(27,36,37,39)(30,35,34,40);(1,2)(3,6)(4,5)(7,8)(9,13,14,22)(10,17,15,21)(11,18,19,24)(12,20,16,23)(25,30)(26,27)(28,36)(29,35)(31,34)(32,40)(33,37)(38,39)Polycyclic presentation:
2^9|a^2=gh,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8843223);