Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| 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 | |
1566 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)(3,4)(5,8)(6,7)(10,14)(11,15)(19,23)(21,24)(25,26,28,30)(27,31,32,29);(1,2,5,8)(3,4,7,6)(9,10)(11,12)(13,14)(15,16)(17,18,20,22)(19,21,23,24);(1,3,5,7)(2,6,8,4)(9,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,30)(29,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,16)(12,15)(13,14)(25,27,28,32)(26,29,30,31);(1,4)(2,7)(3,8)(5,6)(9,12,13,16)(10,11,14,15)(17,19,20,23)(18,21,22,24)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=gi,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4942297);