Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(4,7)(9,10,13,14)(11,15,16,12)(19,20)(23,24)(25,26,29,27)(28,31,32,30);(1,2)(3,7)(4,8)(5,6)(9,11,13,16)(10,12,14,15)(17,18,22,21)(19,24,20,23)(25,27,29,26)(28,31,32,30);(2,6)(4,7)(10,14)(11,16)(17,19,22,20)(18,23,21,24);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,15)(14,16)(17,20,22,19)(18,24,21,23)(25,28)(26,30)(27,31)(29,32);(1,4,5,7)(2,8,6,3)(9,10)(11,12)(13,14)(15,16)(17,21,22,18)(19,24,20,23)Polycyclic presentation:
2^9|a^2=gi,b^2=fghi,c^2=fh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 936893);