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 | |
4430 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,16,21,20)(12,22,17,13)(18,24,23,19)(27,33)(28,36)(31,34)(32,37);(1,3,5,4)(2,6,7,8)(9,11,15,21)(10,16,14,20)(12,18,17,23)(13,19,22,24)(25,26,30,29)(27,34,33,31)(28,32,36,37)(35,40,39,38);(2,7)(6,8)(9,12)(10,13)(11,18)(14,22)(15,17)(16,19)(20,24)(21,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,18)(13,19)(14,20)(15,21)(17,23)(22,24);(1,4,5,3)(2,8,7,6)(9,13,15,22)(10,17,14,12)(11,19,21,24)(16,23,20,18)(25,28,30,36)(26,32,29,37)(27,35,33,39)(31,38,34,40);(1,2)(3,6)(4,8)(5,7)(9,14,15,10)(11,20,21,16)(12,22,17,13)(18,24,23,19)(25,29,30,26)(27,31,33,34)(28,37,36,32)(35,38,39,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,d^2=g,e^2=h,f^2=gh,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, 8087682);