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 | |
4177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,7)(3,5,4,8)(9,10,15,18)(11,16,21,12)(13,14,17,23)(19,20,24,22)(25,26,30,34)(27,31,37,40)(28,29,33,32)(35,36,39,38);(2,7)(3,4)(9,11,15,21)(10,16,18,12)(13,19,17,24)(14,20,23,22)(25,27,30,37)(26,31,34,40)(28,35,33,39)(29,36,32,38);(1,3,6,4)(2,8,7,5)(9,12,15,16)(10,11,18,21)(13,22,17,20)(14,19,23,24)(25,28,30,33)(26,32,34,29)(27,35,37,39)(31,38,40,36);(1,4,6,3)(2,5,7,8)(9,13)(10,14)(11,19)(12,22)(15,17)(16,20)(18,23)(21,24)(25,27)(26,31)(28,35)(29,36)(30,37)(32,38)(33,39)(34,40);(1,5)(2,3)(4,7)(6,8)(9,12)(10,11)(13,22)(14,19)(15,16)(17,20)(18,21)(23,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,14)(10,17)(11,20)(12,19)(13,18)(15,23)(16,24)(21,22)(25,29,30,32)(26,33,34,28)(27,36,37,38)(31,39,40,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8440537);