Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1898 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)(3,5)(4,7)(6,8)(9,10,14,16)(11,12,15,13)(19,23)(20,24)(27,31)(29,32);(2,3)(6,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,24,23,20)(25,26,28,30)(27,29,31,32);(1,2,5,3)(4,6,8,7)(9,12,14,13)(10,15,16,11)(17,19)(18,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32);(1,3,5,2)(4,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,22)(23,24);(1,4,5,8)(2,6,3,7)(9,13,14,12)(10,15,16,11)(17,20)(18,19)(21,24)(22,23)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fh,c^2=f,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4801805);