Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 4194304 |
| 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 | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2872 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)(3,6)(4,7)(5,8)(9,10,13,15)(11,12,16,14)(17,18,20,21)(19,23,24,22)(26,30)(28,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(18,21)(22,23)(25,26)(27,31)(28,32)(29,30);(17,18)(19,22)(20,21)(23,24)(25,27)(26,28)(29,32)(30,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)(17,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,30)(29,32);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19,20,24)(18,22,21,23)(25,28,29,31)(26,32,30,27)Polycyclic presentation:
2^9|a^2=i,d^2=gi,e^2=h,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.
| maximal subgroup of | #72302873, #72302894, #72303515, … |
| maximal normal subgroup of | #72302873, #72302894, #72303515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4802836);