Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(10,14)(11,16)(19,23)(22,24)(25,26,28,30)(27,29,31,32);(2,5)(7,8)(9,10,13,14)(11,15,16,12)(18,21)(22,24)(25,27,28,31)(26,29,30,32);(1,3,6,4)(2,7,5,8)(9,11,13,16)(10,15,14,12)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,31)(30,32);(1,4,6,3)(2,7,5,8)(9,12,13,15)(10,11,14,16)(25,26)(27,29)(28,30)(31,32);(1,5,6,2)(3,7,4,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5393632);