Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,15)(12,13)(14,16)(17,18,21,23)(19,20,24,22)(25,26,29,32)(27,31,28,30);(1,3,4,8)(2,5,7,6)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27,29,28)(26,30,32,31);(3,8)(5,6)(10,12)(14,15)(19,24)(20,22)(25,26,29,32)(27,30,28,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,26)(27,31)(28,30)(29,32);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fgh,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2680926);