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 | |
4639 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,6)(3,4,7,8)(10,15)(11,16)(17,22)(18,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(2,6)(3,7)(9,10,14,15)(11,12,16,19)(13,17,21,22)(18,20,23,24)(26,30)(31,33)(32,36)(39,40);(1,3)(2,4)(5,7)(6,8)(11,16)(12,19)(13,21)(17,22)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,18)(14,16)(15,19)(17,20)(21,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,40)(37,38);(9,12)(10,16)(11,15)(13,20)(14,19)(17,23)(18,22)(21,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,38)(37,39);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,20,19,24)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8930778);