Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,8)(3,6,7,4)(9,10,15,17)(11,18,21,16)(12,13,23,14)(19,22,24,20)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11,15,21)(10,16,17,18)(12,19,23,24)(13,20,14,22)(25,27,30,26)(28,33,36,31)(29,32,38,34)(35,39,37,40);(1,4,5,6)(2,7,8,3)(9,12)(10,13)(11,19)(14,17)(15,23)(16,20)(18,22)(21,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(3,7)(4,6)(9,13,15,14)(10,12,17,23)(11,20,21,22)(16,19,18,24)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40);(2,8)(3,7)(11,21)(16,18)(19,24)(20,22)(26,27)(31,33)(32,34)(39,40);(9,14,15,13)(10,12,17,23)(11,22,21,20)(16,19,18,24)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=h,d^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012764);