Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | #17091 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(10,11)(12,16)(17,18,23,24)(19,20,21,22)(27,31)(29,32);(1,3,6,4)(2,7,8,5)(9,10,13,11)(12,15,16,14)(17,19,23,21)(18,20,24,22)(25,26,28,30)(27,29,31,32);(2,8)(3,4)(9,10,13,11)(12,14,16,15)(17,20,23,22)(18,19,24,21)(25,26)(27,29)(28,30)(31,32);(1,4,6,3)(2,7,8,5)(9,11,13,10)(12,14,16,15)(17,21,23,19)(18,20,24,22)(25,27)(26,29)(28,31)(30,32);(1,5,6,7)(2,3,8,4)(9,12)(10,14)(11,15)(13,16)(17,22,23,20)(18,19,24,21)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=hi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5693620);