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 | |
4488 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)(27,37)(28,34)(29,30)(31,35)(32,36)(33,39)(38,40);(1,2)(3,4)(5,7)(6,8)(11,17)(15,21)(18,23)(22,24)(25,27)(26,33)(28,31)(29,38)(30,35)(32,39)(34,40)(36,37);(1,3)(2,6)(4,7)(5,8)(10,14)(15,21)(16,19)(22,24)(25,28,32,29)(26,34,36,30)(27,31,39,38)(33,40,37,35);(1,3)(2,4)(5,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,29,32,28)(26,34,36,30)(27,38,39,31)(33,40,37,35);(1,4,5,6)(2,3,7,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,30,32,34)(26,29,36,28)(27,35,39,40)(31,33,38,37);(1,3)(2,4)(5,8)(6,7)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,31,32,38)(26,35,36,40)(27,28,39,29)(30,37,34,33)Polycyclic presentation:
2^9|c^2=g,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10051788);