Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2905 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,16,12,14)(18,20)(19,24)(25,26)(27,28)(29,31)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,18,21,20)(19,23,24,22)(25,27)(26,30)(28,31)(29,32);(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,20,21,18)(19,23,24,22)(25,28,29,30)(26,27,31,32);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,14,15,16)(17,19)(18,23)(20,22)(21,24)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fg,b^2=i,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4331075);