Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 |
| prime radical | #34964 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4422 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,20)(14,17)(15,18)(16,22)(19,23)(21,24)(25,26)(27,30)(28,35)(29,31)(32,36)(33,38)(34,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11,15,17)(10,13,18,14)(12,19,22,24)(16,21,20,23)(25,27,29,33)(26,30,31,38)(28,34,36,39)(32,37,35,40);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,24)(18,20)(26,31)(27,33)(32,35)(34,39);(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,24,20,19)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(1,4,5,3)(2,6,7,8)(12,22)(16,20)(19,24)(21,23);(9,14,15,13)(10,17,18,11)(12,23,22,21)(16,24,20,19)Polycyclic presentation:
2^9|b^2=gh,d^2=h,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^b=dhi,e^c=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7548999);