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 | |
4481 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)(3,6)(4,7)(5,8)(9,10)(11,19)(12,13)(14,17)(15,18)(16,22)(20,23)(21,24)(25,26,30,33)(27,29,35,31)(28,36,38,32)(34,37,40,39);(1,3,6,2)(4,7,5,8)(9,11,15,22)(10,16,18,19)(12,20,17,24)(13,21,14,23)(25,27,30,35)(26,31,33,29)(28,34,38,40)(32,37,36,39);(1,4,6,5)(2,7,3,8)(9,12,15,17)(10,13,18,14)(11,20,22,24)(16,21,19,23)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,39,31,37);(1,5,6,4)(2,7,3,8)(9,13,15,14)(10,17,18,12)(11,21,22,23)(16,24,19,20)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,40,36,34);(11,22)(16,19)(20,24)(21,23)(26,33)(27,35)(32,36)(34,40);(9,14,15,13)(10,17,18,12)(11,23,22,21)(16,24,19,20)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,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, 8012691);