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 | |
4305 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,15,13)(11,18,16,12)(14,17,22,21)(19,24,23,20)(25,26,31,33)(27,36,30,32)(28,38,39,34)(29,35,40,37);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,27,31,30)(26,32,33,36)(28,37,39,35)(29,38,40,34);(1,3,5,4)(2,6,7,8)(26,33)(28,39)(32,36)(35,37);(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,28,31,39)(26,34,33,38)(27,37,30,35)(29,36,40,32);(1,4,5,3)(2,6,7,8)(9,13,15,10)(11,18,16,12)(14,17,22,21)(19,20,23,24)(25,29)(26,35)(27,34)(28,36)(30,38)(31,40)(32,39)(33,37);(1,2)(3,6)(4,8)(5,7)(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,30,31,27)(26,36,33,32)(28,35,39,37)(29,34,40,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9613982);