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 | |
4444 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)(4,5)(6,8)(10,15)(11,16)(17,22)(18,23);(3,8)(6,7)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(25,26,31,35)(27,36,38,32)(28,30,34,40)(29,33,37,39);(1,3,5,8)(2,6,4,7)(11,16)(12,19)(13,21)(17,22)(25,27,31,38)(26,32,35,36)(28,29,34,37)(30,39,40,33);(3,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,28,31,34)(26,30,35,40)(27,37,38,29)(32,33,36,39);(1,3)(2,7)(4,6)(5,8)(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,29,31,37)(26,33,35,39)(27,28,38,34)(30,32,40,36);(1,4,5,2)(3,6,8,7)(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,37)(38,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=i,d^2=hi,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8924622);