Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | #16250 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,21)(22,24)(25,26,28,30)(27,29,32,31);(1,3)(2,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(18,21)(19,24)(25,26)(27,31)(28,30)(29,32);(2,8)(3,4)(11,15)(13,16)(17,18,20,21)(19,22,24,23)(25,26,28,30)(27,29,32,31);(1,4,6,3)(2,5,8,7)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,29)(28,30)(31,32);(1,5)(2,3)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,24)(18,22,21,23)(25,27)(26,29)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gi,d^2=fgi,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1700277);