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 | |
4157 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
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)(27,35)(28,31)(29,33)(30,38)(32,36)(34,39)(37,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,12)(13,18)(14,19)(15,16)(17,20)(21,24)(22,23)(25,27,31,36)(26,32,28,35)(29,39,40,30)(33,38,37,34);(1,3,5,4)(2,6,7,8)(26,28)(32,35)(33,37)(34,38);(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,28,31,26)(27,35,36,32)(29,37,40,33)(30,38,39,34);(1,4,5,3)(2,6,7,8)(9,13,15,10)(11,18,16,12)(14,17,22,21)(19,20,23,24)(25,29,31,40)(26,33,28,37)(27,30,36,39)(32,34,35,38);(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)(26,34)(27,29)(28,38)(31,39)(32,33)(35,37)(36,40)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=i,d^2=g,e^2=gi,f^2=gh,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, 9614351);