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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2024 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,16,12,15)(19,23)(21,24)(27,31)(30,32);(1,3,5,4)(2,6,7,8)(10,14)(15,16)(17,18,20,22)(19,21,23,24)(26,29)(30,32);(3,4)(6,8)(9,11,13,12)(10,15,14,16)(17,19,20,23)(18,21,22,24)(25,26,28,29)(27,30,31,32);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,15,14,16)(17,19)(18,21)(20,23)(22,24);(1,2)(3,6)(4,8)(5,7)(9,10)(11,15)(12,16)(13,14)(17,18)(19,21)(20,22)(23,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fhi,d^2=f,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, 5626385);