Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10,12,13)(11,15,16,14)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,31,32,28);(2,5)(3,7)(10,13)(14,15)(17,18,20,21)(19,23,24,22)(26,30)(28,31);(2,5)(6,8)(11,16)(14,15)(18,21)(22,23)(25,27,29,32)(26,31,30,28);(1,2)(3,8)(4,5)(6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,30)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=fhi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3165182);