Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | #52280 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4392 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)(4,5)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(2,5)(6,8)(27,33)(30,37)(34,35)(38,39);(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,29,33)(26,30,32,37)(28,35,36,34)(31,39,40,38);(10,15)(11,18)(16,20)(19,24)(25,28,29,36)(26,31,32,40)(27,34,33,35)(30,38,37,39);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(26,32)(30,37)(31,40)(38,39);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541794);