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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1701 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10,13,16)(11,15,12,14)(17,18,20,21)(19,23,24,22)(25,26,30,28)(27,32,31,29);(1,2,5,3)(4,7,8,6)(18,21)(22,23);(9,11,13,12)(10,14,16,15)(17,18)(19,22)(20,21)(23,24)(25,27)(26,29)(28,32)(30,31);(1,3,5,2)(4,6,8,7)(9,10)(11,14)(12,15)(13,16)(17,19,20,24)(18,22,21,23)(25,28,30,26)(27,32,31,29);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,15,16,14)(17,18)(19,23)(20,21)(22,24)(25,29,30,32)(26,31,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=fi,d^2=ghi,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, 1406740);