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 | |
4382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,8)(3,4,5,7)(25,26)(27,28)(29,39)(30,33)(31,34)(32,38)(35,40)(36,37);(1,3,6,5)(2,7,8,4)(9,10,14,15)(11,13,17,22)(12,16,20,23)(18,21,24,19)(25,27,31,32)(26,28,34,38)(29,35,33,37)(30,36,39,40);(1,3,6,5)(2,7,8,4)(10,15)(11,17)(16,23)(18,24)(25,28,31,38)(26,32,34,27)(29,36,33,40)(30,37,39,35);(2,8)(3,5)(9,11)(10,13)(12,18)(14,17)(15,22)(16,21)(19,23)(20,24)(25,28)(26,27)(29,36)(30,35)(31,38)(32,34)(33,40)(37,39);(1,4)(2,3)(5,8)(6,7)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,29)(26,30)(27,35)(28,36)(31,33)(32,37)(34,39)(38,40);(1,5,6,3)(2,7,8,4)(9,13)(10,17)(11,15)(12,19)(14,22)(16,18)(20,21)(23,24)(25,30)(26,33)(27,36)(28,37)(29,34)(31,39)(32,40)(35,38)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8616325);