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 | |
4697 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)(9,10,14,15)(11,19,20,16)(12,17,22,21)(13,23,24,18)(25,26)(27,36)(28,30)(29,33)(31,35)(32,37)(34,40)(38,39);(1,2,4,6)(3,5,7,8)(10,15)(12,22)(16,19)(18,23)(27,37)(29,38)(32,36)(33,39);(1,3)(2,5)(4,7)(6,8)(10,15)(11,20)(12,22)(13,24)(25,27,31,37)(26,32,35,36)(28,29,34,38)(30,39,40,33);(9,11,14,20)(10,16,15,19)(12,23,22,18)(13,21,24,17)(25,28,31,34)(26,30,35,40)(27,38,37,29)(32,33,36,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,21)(16,24)(20,23)(25,29,31,38)(26,33,35,39)(27,28,37,34)(30,32,40,36);(1,3)(2,5)(4,7)(6,8)(9,13,14,24)(10,18,15,23)(11,21,20,17)(12,19,22,16)(25,30)(26,34)(27,33)(28,35)(29,32)(31,40)(36,38)(37,39)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=i,d^2=gi,e^2=i,f^2=g,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, 8922879);