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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2432 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)(3,8)(4,6)(5,7)(9,10,12,13)(11,15,16,14)(18,21)(22,23)(26,29)(30,32);(1,3,5,6)(2,4,7,8)(10,13)(14,15)(17,18)(19,23)(20,21)(22,24)(27,31)(30,32);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,30)(28,29)(31,32);(1,2)(3,4)(5,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,23)(20,24)(21,22)(25,27,28,31)(26,30,29,32);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fh,d^2=gi,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 | #72266502, #72266510, #72266511, … |
| maximal normal subgroup of | #72266502, #72266510, #72266511, … |
| maximal quotient of | #72334798 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4801918);