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 | |
4464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,7)(9,10,15,18)(11,19,21,16)(12,22,23,13)(14,17,20,24)(25,26)(27,33)(28,32)(29,30)(31,34)(35,40)(36,39)(37,38);(1,2,5,4)(3,7,8,6)(9,11,15,21)(10,16,18,19)(12,17,23,24)(13,20,22,14)(27,34)(31,33)(35,36)(39,40);(1,3,5,8)(2,6,4,7)(12,23)(13,22)(14,20)(17,24)(25,27,30,34)(26,31,29,33)(28,36,38,35)(32,40,37,39);(1,4,5,2)(3,7,8,6)(9,12,15,23)(10,13,18,22)(11,17,21,24)(14,16,20,19)(25,28,30,38)(26,32,29,37)(27,35,34,36)(31,39,33,40);(9,13)(10,12)(11,20)(14,21)(15,22)(16,17)(18,23)(19,24)(25,27)(26,33)(28,36)(29,31)(30,34)(32,39)(35,38)(37,40);(9,14,15,20)(10,17,18,24)(11,13,21,22)(12,16,23,19)(25,29,30,26)(27,31,34,33)(28,37,38,32)(35,39,36,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=i,d^2=gh,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, 8933947);