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 | |
4495 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)(3,4)(5,7)(6,8)(9,10)(11,16)(12,18)(13,19)(14,23)(15,17)(20,21)(22,24)(25,26)(27,30)(28,35)(29,32)(31,36)(33,37)(34,40)(38,39);(1,3,6,5)(2,7,8,4)(9,11,15,19)(10,16,17,13)(12,21,22,14)(18,20,24,23)(27,33)(30,37)(34,39)(38,40);(10,17)(11,19)(12,22)(20,23)(25,26,29,32)(27,30,33,37)(28,31,36,35)(34,38,39,40);(1,4,6,7)(2,5,8,3)(9,12)(10,18)(11,21)(13,23)(14,19)(15,22)(16,20)(17,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,5,6,3)(2,7,8,4)(9,13)(10,19)(11,17)(12,20)(14,24)(15,16)(18,21)(22,23)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(1,2)(3,7)(4,5)(6,8)(9,14,15,21)(10,20,17,23)(11,12,19,22)(13,18,16,24)(25,28)(26,31)(27,34)(29,36)(30,38)(32,35)(33,39)(37,40)Polycyclic presentation:
2^9|b^2=gi,c^2=hi,d^2=g,e^2=g,f^2=i,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, 10103275);