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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1658 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,14,12)(11,16,15,13)(17,18)(19,21)(20,22)(23,24);(2,5)(6,7)(9,11,14,15)(10,13,12,16)(18,19)(20,24)(26,29)(30,32);(3,8)(6,7)(11,15)(13,16)(17,18,21,19)(20,23,24,22)(25,26)(27,30)(28,29)(31,32);(1,2)(3,7)(4,5)(6,8)(9,12,14,10)(11,13,15,16)(17,19,21,18)(20,22,24,23)(25,27,28,31)(26,30,29,32);(1,3)(2,6)(4,8)(5,7)(9,13)(10,15)(11,12)(14,16)(17,20,21,24)(18,22,19,23)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=g,d^2=h,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, 4035639);