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 | |
4481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,19)(12,23)(13,24)(14,18)(15,22)(16,21)(17,20)(25,26,31,36)(27,37,39,32)(28,29,34,33)(30,35,40,38);(2,5)(6,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,19)(23,24)(25,27,31,39)(26,32,36,37)(28,30,34,40)(29,38,33,35);(1,2)(3,6)(4,5)(7,8)(25,28,31,34)(26,33,36,29)(27,30,39,40)(32,35,37,38);(1,2)(3,6)(4,5)(7,8)(9,12,14,21)(10,16,18,23)(11,20,22,13)(15,17,19,24)(25,26)(27,37)(28,33)(29,34)(30,38)(31,36)(32,39)(35,40);(1,3,4,8)(2,6,5,7)(9,13,14,20)(10,17,18,24)(11,21,22,12)(15,16,19,23)(25,29,31,33)(26,34,36,28)(27,38,39,35)(30,32,40,37);(9,10)(11,15)(12,23)(13,17)(14,18)(16,21)(19,22)(20,24)(25,30)(26,35)(27,34)(28,39)(29,32)(31,40)(33,37)(36,38)Polycyclic presentation:
2^9|a^2=h,b^2=hi,c^2=hi,d^2=gh,e^2=g,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, 8343813);