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 | |
4605 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26)(27,30)(28,31)(29,34)(32,37)(33,36)(35,39)(38,40);(1,2,5,3)(4,7,8,6)(11,17)(15,21)(18,23)(22,24)(25,26,30,27)(28,33,36,31)(29,32,37,34)(35,39,40,38);(2,3)(6,7)(10,14)(15,21)(16,19)(22,24)(26,27)(31,33)(32,34)(38,39);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,5,2)(4,7,8,6)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(1,4,5,8)(2,6,3,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|b^2=i,d^2=g,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9253890);