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 | |
4388 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)(3,6)(4,7)(5,8)(9,10,15,20)(11,21,23,16)(12,13,18,17)(14,19,22,24)(25,26)(27,36)(28,33)(29,37)(30,38)(31,34)(32,35)(39,40);(3,8)(5,6)(9,11)(10,16)(12,22)(13,19)(14,18)(15,23)(17,24)(20,21)(25,27,32,34)(26,31,35,36)(28,37,39,30)(29,33,38,40);(1,3)(2,5)(4,8)(6,7)(9,12,15,18)(10,17,20,13)(11,22,23,14)(16,24,21,19)(25,28,32,39)(26,33,35,40)(27,37,34,30)(29,31,38,36);(1,3)(2,6)(4,8)(5,7)(9,10)(11,21)(12,17)(13,18)(14,24)(15,20)(16,23)(19,22)(25,29)(26,30)(27,33)(28,31)(32,38)(34,40)(35,37)(36,39);(9,13,15,17)(10,18,20,12)(11,19,23,24)(14,21,22,16)(25,30)(26,29)(27,28)(31,33)(32,37)(34,39)(35,38)(36,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,22)(10,19,20,24)(11,18,23,12)(13,21,17,16)(25,31)(26,34)(27,35)(28,38)(29,39)(30,33)(32,36)(37,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=g,e^2=hi,f^2=hi,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, 8370892);