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 | |
4245 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)(3,4)(5,6)(7,8)(9,10)(11,12)(13,21)(14,15)(16,23)(17,18)(19,22)(20,24)(25,26)(27,37)(28,33)(29,39)(30,34)(31,35)(32,36)(38,40);(2,6)(3,7)(9,11)(10,12)(13,19)(14,17)(15,18)(16,20)(21,22)(23,24)(25,27,32,35)(26,31,36,37)(28,30,29,40)(33,38,39,34);(1,3,5,7)(2,4,6,8)(10,15)(11,17)(16,21)(19,24)(25,28,32,29)(26,33,36,39)(27,30,35,40)(31,38,37,34);(1,4)(2,7)(3,6)(5,8)(9,10)(11,18)(12,17)(13,16)(14,15)(19,22)(20,24)(21,23)(25,29,32,28)(26,33,36,39)(27,30,35,40)(31,34,37,38);(9,12,14,18)(10,11,15,17)(13,22,23,20)(16,24,21,19)(25,30)(26,34)(27,29)(28,35)(31,33)(32,40)(36,38)(37,39);(9,13)(10,16)(11,19)(12,20)(14,23)(15,21)(17,24)(18,22)(25,31)(26,35)(27,36)(28,38)(29,34)(30,39)(32,37)(33,40)Polycyclic presentation:
2^9|b^2=ghi,c^2=g,d^2=ghi,e^2=h,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, 10154342);