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 | |
4445 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)(3,4)(5,7)(6,8)(9,10)(11,20)(12,15)(13,23)(14,18)(16,19)(17,22)(21,24)(25,26,31,32)(27,36,37,33)(28,40,34,29)(30,35,38,39);(1,3)(2,6)(4,7)(5,8)(9,11,16,22)(10,17,19,20)(12,18,13,24)(14,15,21,23)(26,32)(27,37)(28,34)(35,39);(2,7)(4,6)(9,12,16,13)(10,15,19,23)(11,18,22,24)(14,17,21,20)(25,27)(26,33)(28,38)(29,39)(30,34)(31,37)(32,36)(35,40);(1,3)(2,6)(4,7)(5,8)(9,13,16,12)(10,15,19,23)(11,18,22,24)(14,20,21,17)(25,28,31,34)(26,29,32,40)(27,38,37,30)(33,39,36,35);(1,3)(2,6)(4,7)(5,8)(9,14,16,21)(10,18,19,24)(11,15,22,23)(12,17,13,20)(25,29)(26,34)(27,39)(28,32)(30,33)(31,40)(35,37)(36,38);(1,4,5,6)(2,3,7,8)(9,15)(10,12)(11,21)(13,19)(14,22)(16,23)(17,18)(20,24)(25,30,31,38)(26,35,32,39)(27,34,37,28)(29,33,40,36)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=ghi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8095240);