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 | |
4364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,16)(12,18)(13,19)(14,21)(15,17)(20,24)(22,23)(26,31)(27,34)(29,37)(36,39);(1,2)(3,5)(4,7)(6,8)(9,11,15,12)(10,16,17,18)(13,20,23,21)(14,19,24,22)(25,26,30,31)(27,28,34,32)(29,33,37,38)(35,36,40,39);(1,3)(2,5)(4,8)(6,7)(10,17)(11,12)(19,22)(20,21)(25,27,30,34)(26,32,31,28)(29,35,37,40)(33,39,38,36);(1,3,4,8)(2,5,7,6)(9,12,15,11)(10,18,17,16)(13,21,23,20)(14,22,24,19)(25,27)(26,28)(29,35)(30,34)(31,32)(33,36)(37,40)(38,39);(1,3)(2,6)(4,8)(5,7)(9,13,15,23)(10,19,17,22)(11,21,12,20)(14,18,24,16)(25,28)(26,27)(29,36)(30,32)(31,34)(33,35)(37,39)(38,40);(1,2)(3,5)(4,7)(6,8)(9,14,15,24)(10,20,17,21)(11,19,12,22)(13,16,23,18)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=ghi,d^2=gi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10050494);