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 | |
4472 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)(11,14)(12,15)(13,18)(16,22)(17,19)(20,24)(21,23)(25,26)(27,30)(28,35)(29,31)(32,36)(33,39)(34,37)(38,40);(1,2,5,6)(3,7,8,4)(10,11)(12,19)(16,18)(20,21)(25,27,29,34)(26,30,31,37)(28,33,36,40)(32,38,35,39);(1,3)(2,4)(5,8)(6,7)(10,11)(15,17)(16,18)(23,24)(26,31)(30,37)(32,35)(38,39);(2,6)(4,7)(9,11,14,10)(12,15,19,17)(13,18,22,16)(20,23,21,24)(25,26)(27,30)(28,32)(29,31)(33,38)(34,37)(35,36)(39,40);(1,4)(2,3)(5,7)(6,8)(9,12,14,19)(10,15,11,17)(13,21,22,20)(16,24,18,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,3)(2,7)(4,6)(5,8)(9,13,14,22)(10,16,11,18)(12,20,19,21)(15,23,17,24)(25,28)(26,32)(27,33)(29,36)(30,38)(31,35)(34,40)(37,39)Polycyclic presentation:
2^9|b^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10132760);