Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #17082 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(4,8)(9,10)(11,14)(12,13)(15,16)(18,22)(19,24)(25,26)(27,28)(29,30)(31,32);(1,2,5,3)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,18,21,22)(19,23,24,20)(25,26)(27,28)(29,30)(31,32);(4,8)(6,7)(10,12)(14,15)(18,22)(20,23)(26,30)(27,31);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,15,16,14)(17,19,21,24)(18,23,22,20)(25,27,29,31)(26,28,30,32);(1,4,5,8)(2,6,3,7)(9,11)(10,14)(12,15)(13,16)(17,20,21,23)(18,24,22,19)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|b^2=i,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389377);