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) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,20,23,22)(25,26)(27,29)(28,30)(31,32);(1,2)(3,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,22)(20,24)(21,23)(25,26,28,30)(27,31,32,29);(3,8)(5,7)(10,14)(11,16)(19,23)(20,22)(25,26)(27,31)(28,30)(29,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,24);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,20,21,22)(18,23,24,19)(25,27,28,32)(26,29,30,31)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,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, 5393619);