Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
1702 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,20)(21,24)(22,23)(25,26,30,28)(27,32,29,31);(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,20,23,24)(26,28)(31,32);(2,3)(6,7)(10,11)(14,15)(18,23)(20,24)(25,27,30,29)(26,31,28,32);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,20,22,24)(18,19,23,21)(25,28,30,26)(27,31,29,32);(1,4,5,8)(2,6,3,7)(9,12)(10,14)(11,15)(13,16)(17,21,22,19)(18,24,23,20)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=fh,b^2=fgi,c^2=f,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4837542);