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×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
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)(18,22)(20,23)(25,26)(27,28)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,18,21,22)(19,20,24,23)(25,27)(26,30)(28,32)(29,31);(17,19)(18,23)(20,22)(21,24)(25,28)(26,31)(27,32)(29,30);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32);(1,2)(3,5)(4,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,22)(21,23)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=h,d^2=i,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 | #72232641, #72232643, #72232674, … |
| maximal normal subgroup of | #72232641, #72232643, #72232674, … |
| maximal quotient of | #72251676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4801792);