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 | |
4505 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)(4,6)(5,8)(9,10)(11,15)(12,20)(13,17)(14,19)(16,21)(18,22)(23,24)(25,26,30,35)(27,34,37,31)(28,38,39,32)(29,33,36,40);(1,2,4,6)(3,5,8,7)(25,27)(26,31)(28,36)(29,39)(30,37)(32,40)(33,38)(34,35);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,23)(21,24)(25,28,30,39)(26,32,35,38)(27,36,37,29)(31,40,34,33);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,23)(20,22)(21,24)(25,29)(26,33)(27,28)(30,36)(31,32)(34,38)(35,40)(37,39);(9,10)(11,14)(12,16)(13,17)(15,19)(18,22)(20,21)(23,24)(25,27)(26,34)(28,36)(29,39)(30,37)(31,35)(32,33)(38,40);(9,12,13,21)(10,16,17,20)(11,18,19,24)(14,22,15,23)(25,26)(27,31)(28,32)(29,33)(30,35)(34,37)(36,40)(38,39)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441509);