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 | #52169 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4372 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,14,18,13)(12,20,22,16)(19,23,24,21)(25,26,29,31)(27,30,33,38)(28,35,36,32)(34,40,39,37);(1,3,5,4)(2,6,8,7)(9,11,15,18)(10,13,17,14)(12,19,22,24)(16,23,20,21)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,12)(10,16)(11,19)(13,23)(14,21)(15,22)(17,20)(18,24)(26,31)(27,33)(32,35)(34,39);(1,3,5,4)(2,7,8,6)(10,17)(11,18)(16,20)(19,24)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(1,4,5,3)(2,6,8,7)(9,13,15,14)(10,11,17,18)(12,21,22,23)(16,24,20,19);(9,14,15,13)(10,18,17,11)(12,21,22,23)(16,24,20,19)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=gh,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, 7548992);