Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | C24 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,25)(20,24)(21,23)(22,26)(27,31)(28,30)(29,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15);(9,10,12,13)(11,15,16,14)(17,19)(18,22)(20,27)(21,26)(23,25)(24,29)(28,32)(30,31);(18,23)(22,25)(24,30)(29,31);(9,11,12,16)(10,14,13,15)(19,26)(22,25)(27,32)(29,31);(17,20,21,28)(18,24,23,30)(19,27,26,32)(22,29,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7544333);