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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,16)(12,13)(14,19)(15,17)(18,24)(20,21)(22,23)(25,26,31,32)(27,29,35,39)(28,37,36,33)(30,34,40,38);(1,3,5,8)(2,6,4,7)(9,11,15,14)(10,16,17,19)(12,20,18,22)(13,21,24,23)(25,27,31,35)(26,29,32,39)(28,30,36,40)(33,38,37,34);(10,17)(12,18)(16,19)(20,22)(26,32)(27,35)(28,36)(34,38);(9,12,15,18)(10,13,17,24)(11,20,14,22)(16,21,19,23)(25,28,31,36)(26,33,32,37)(27,30,35,40)(29,38,39,34);(1,4,5,2)(3,7,8,6)(9,13)(10,18)(11,21)(12,17)(14,23)(15,24)(16,22)(19,20)(25,29,31,39)(26,27,32,35)(28,38,36,34)(30,37,40,33);(9,14,15,11)(10,19,17,16)(12,22,18,20)(13,23,24,21)(25,30)(26,34)(27,36)(28,35)(29,33)(31,40)(32,38)(37,39)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=h,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7834330);