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 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,21)(14,15)(16,18)(17,23)(19,22)(20,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(3,8)(6,7)(10,15)(12,16)(17,21)(20,22)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(1,3)(2,6)(4,7)(5,8)(10,15)(11,18)(17,21)(19,24)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(3,8)(6,7)(9,11,14,18)(10,16,15,12)(13,19,23,24)(17,22,21,20)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,3)(2,7)(4,6)(5,8)(9,12,14,16)(10,11,15,18)(13,22,23,20)(17,24,21,19)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,4,5,2)(3,7,8,6)(9,13,14,23)(10,17,15,21)(11,19,18,24)(12,20,16,22)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39)Polycyclic presentation:
2^9|b^2=ghi,c^2=ghi,d^2=g,e^2=g,f^2=gi,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, 10103345);