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 | |
4709 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)(5,6)(7,8)(9,10,14,17)(11,15,19,13)(12,16,21,24)(18,20,23,22)(25,26,29,32)(27,33,34,30)(28,31,38,37)(35,40,36,39);(2,6)(3,7)(9,11,14,19)(10,13,17,15)(12,18,21,23)(16,22,24,20)(27,34)(30,33)(35,36)(39,40);(1,3,5,7)(2,4,6,8)(25,27)(26,30)(28,35)(29,34)(31,39)(32,33)(36,38)(37,40);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,19)(16,20)(21,23)(22,24)(25,27)(26,30)(28,36)(29,34)(31,40)(32,33)(35,38)(37,39);(9,12)(10,16)(11,18)(13,22)(14,21)(15,20)(17,24)(19,23)(25,28)(26,31)(27,35)(29,38)(30,39)(32,37)(33,40)(34,36);(9,13,14,15)(10,11,17,19)(12,20,21,22)(16,23,24,18)(25,26)(27,30)(28,37)(29,32)(31,38)(33,34)(35,40)(36,39)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10441571);