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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1556 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)(3,4)(5,7)(6,8)(9,10,13,12)(11,15,16,14)(19,23)(20,24);(1,3,5,8)(2,6,7,4)(17,18,21,22)(19,24,23,20)(25,26,29,32)(27,30,28,31);(1,4,5,6)(2,3,7,8)(9,10,13,12)(11,14,16,15)(17,19)(18,20)(21,23)(22,24)(25,26)(27,30)(28,31)(29,32);(1,4)(2,3)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,20)(21,22)(23,24)(25,27,29,28)(26,30,32,31);(1,2)(3,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,24)(22,23)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fi,d^2=fghi,e^2=fghi,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, 1695992);