Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | #16260 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10,13,12)(11,14,15,16)(17,18,21,22)(19,20,24,23)(25,26,28,30)(27,29,31,32);(10,12)(14,16)(17,18,21,22)(19,23,24,20)(27,31)(29,32);(2,5)(6,7)(11,15)(14,16)(18,22)(19,24)(25,27,28,31)(26,29,30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=gi,c^2=ghi,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5959112);