Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | #15273 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2466 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)(3,7)(4,8)(5,6)(10,13)(11,15)(18,19)(20,24)(26,27)(30,31);(2,6)(4,7)(9,10,12,13)(11,16,15,14)(18,19)(22,23)(25,26,29,27)(28,31,32,30);(2,6)(4,7)(11,15)(14,16)(17,18,21,19)(20,22,24,23)(25,26,29,27)(28,31,32,30);(1,3)(2,7)(4,6)(5,8)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,23,24,22)(25,27,29,26)(28,30,32,31);(1,4,5,7)(2,8,6,3)(9,11,12,15)(10,14,13,16)(17,20)(18,22)(19,23)(21,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|b^2=hi,c^2=fghi,d^2=fghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3940188);