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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,12,16,14)(18,19)(20,24)(25,26,29,31)(27,28,32,30);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,15,12)(18,19)(22,23)(25,27,29,32)(26,30,31,28);(3,4)(6,7)(10,15)(12,14)(17,18,21,19)(20,22,24,23)(25,26,29,31)(27,30,32,28);(1,4,5,3)(2,7,8,6)(9,12,13,14)(10,11,15,16)(17,19,21,18)(20,23,24,22)(25,26)(27,30)(28,32)(29,31);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,16)(14,15)(17,20,21,24)(18,22,19,23)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=fghi,e^2=ghi,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, 3236724);