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 | |
4599 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,19)(12,17)(13,20)(14,15)(16,21)(18,23)(22,24)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(1,3,6,5)(2,7,4,8)(10,15)(12,22)(16,19)(18,20)(27,29)(31,34)(35,36)(38,39);(9,11,14,21)(10,16,15,19)(12,20,22,18)(13,17,23,24)(25,27,30,29)(26,31,33,34)(28,36,37,35)(32,39,40,38);(2,4)(7,8)(9,11)(10,16)(12,18)(13,17)(14,21)(15,19)(20,22)(23,24)(25,28,30,37)(26,32,33,40)(27,35,29,36)(31,38,34,39);(1,4,6,2)(3,8,5,7)(9,12,14,22)(10,17,15,24)(11,20,21,18)(13,16,23,19)(25,29,30,27)(26,31,33,34)(28,35,37,36)(32,39,40,38);(1,5,6,3)(2,7,4,8)(9,13,14,23)(10,18,15,20)(11,17,21,24)(12,19,22,16)(25,26)(27,31)(28,32)(29,34)(30,33)(35,38)(36,39)(37,40)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=h,d^2=i,e^2=gi,f^2=g,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, 10078832);