Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | #16284 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,8,4)(9,10,15,14)(11,12,16,13)(18,22)(20,24)(27,31)(30,32);(2,6)(3,8)(9,11)(10,13)(12,14)(15,16)(18,22)(19,23)(26,29)(30,32);(2,6)(4,7)(9,12,15,13)(10,16,14,11)(17,18)(19,20)(21,22)(23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,7,6,4)(9,13,15,12)(10,16,14,11)(17,19)(18,20)(21,23)(22,24);(1,4,5,7)(2,8,6,3)(9,14,15,10)(11,12,16,13)(17,20)(18,19)(21,24)(22,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4538485);