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 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,14)(11,15,16,12)(18,22)(20,23)(25,26,29,32)(27,28,31,30);(2,4)(3,8)(9,11,13,16)(10,12,14,15)(17,18,21,22)(19,23,24,20)(25,27,29,31)(26,30,32,28);(2,4)(6,7)(10,14)(11,16)(18,22)(20,23)(25,28,29,30)(26,27,32,31);(1,3,5,8)(2,6,4,7)(9,12)(10,11)(13,15)(14,16)(17,19,21,24)(18,20,22,23)(25,26)(27,28)(29,32)(30,31);(1,4,5,2)(3,6,8,7)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fhi,d^2=gh,e^2=fh,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, 4038412);