Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| automorphism | Aut(G) order 1048576 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10)(11,12)(13,15)(14,16)(18,19)(20,24)(27,29)(31,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,15)(12,16)(13,14)(20,24)(22,23)(25,26,30,28)(27,32,29,31);(2,5)(6,7)(10,16)(11,13)(17,18,21,19)(20,22,24,23)(25,27,30,29)(26,31,28,32);(1,3,4,8)(2,6,5,7)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,23,24,22)(25,28,30,26)(27,32,29,31);(1,2)(3,7)(4,5)(6,8)(9,13,14,11)(10,15,16,12)(17,20,21,24)(18,22,19,23)(25,29,30,27)(26,32,28,31)Polycyclic presentation:
2^9|b^2=fh,c^2=fhi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4038132);