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 | |
4731 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)(9,10)(11,12)(13,21)(14,15)(16,18)(17,22)(19,23)(20,24)(26,27)(31,33)(32,34)(38,39);(3,8)(6,7)(10,15)(12,16)(17,21)(20,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(1,3)(2,6)(4,7)(5,8)(10,15)(11,18)(13,22)(20,23)(29,37)(32,34)(35,40)(38,39);(3,8)(6,7)(9,11,14,18)(10,16,15,12)(13,19,22,24)(17,23,21,20)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,11,15,18)(13,20,22,23)(17,19,21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,2)(3,6,8,7)(9,13,14,22)(10,17,15,21)(11,19,18,24)(12,20,16,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)Polycyclic presentation:
2^9|b^2=ghi,d^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,d^c=dh,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9347764);