Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(10,14)(11,15)(18,19)(20,24)(25,26,31,32)(27,29,30,28);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,15,12)(18,19)(22,23)(25,27,31,30)(26,28,32,29);(2,7)(6,8)(10,14)(12,16)(17,18,21,19)(20,22,24,23)(25,28,31,29)(26,27,32,30);(1,4,5,3)(2,8,7,6)(9,11)(10,12)(13,15)(14,16)(17,19,21,18)(20,23,24,22)(25,29,31,28)(26,27,32,30);(1,2)(3,6)(4,8)(5,7)(9,12)(10,11)(13,16)(14,15)(17,20,21,24)(18,22,19,23)(25,30,31,27)(26,28,32,29)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=ghi,d^2=fh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2456227);