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 | |
4352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(10,12)(11,17)(16,20)(19,21)(25,26,31,35)(27,36,37,32)(28,33,39,38)(29,30,34,40);(1,3,5,8)(2,6,7,4)(9,10,14,12)(11,15,17,18)(13,16,22,20)(19,23,21,24)(27,37)(28,39)(32,36)(33,38);(1,3)(2,4)(5,8)(6,7)(25,27,31,37)(26,32,35,36)(28,34,39,29)(30,38,40,33);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,18)(13,21)(14,17)(16,24)(19,22)(20,23)(25,28)(26,33)(27,34)(29,37)(30,32)(31,39)(35,38)(36,40);(9,12,14,10)(11,18,17,15)(13,20,22,16)(19,24,21,23)(25,29,31,34)(26,30,35,40)(27,28,37,39)(32,38,36,33);(9,13,14,22)(10,16,12,20)(11,19,17,21)(15,23,18,24)(25,30)(26,34)(27,38)(28,36)(29,35)(31,40)(32,39)(33,37)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=h,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8367867);