Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,5)(3,8,4,7)(9,10,14,13)(11,12,16,15)(17,18,22,24)(19,20,23,21)(25,26,29,32)(27,30,31,28);(1,3,6,4)(2,7,5,8)(9,11,14,16)(10,15,13,12)(17,19,22,23)(18,20,24,21)(25,27,29,31)(26,30,32,28);(9,12)(10,16)(11,13)(14,15)(17,20,22,21)(18,19,24,23)(25,26)(27,30)(28,31)(29,32);(1,4,6,3)(2,8,5,7)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24);(1,5,6,2)(3,7,4,8)(9,13,14,10)(11,15,16,12)(17,21,22,20)(18,23,24,19)(25,28,29,30)(26,31,32,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fi,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1372809);