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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(11,12)(14,15)(18,22)(20,24)(25,26,29,31)(27,32,30,28);(3,7)(6,8)(9,10,13,16)(11,14,12,15)(18,22)(19,23)(25,27,29,30)(26,28,31,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,18)(19,20)(21,22)(23,24)(25,27)(26,28)(29,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,20)(18,19)(21,24)(22,23)(25,28)(26,30)(27,31)(29,32)Polycyclic presentation:
2^9|a^2=f,b^2=fghi,c^2=ghi,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, 2462460);