Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2645 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(9,10)(11,12)(13,15)(14,16)(18,22)(19,24)(25,26,29,32)(27,28,31,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,22)(23,24)(25,27)(26,30)(28,32)(29,31);(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,28,29,30)(26,31,32,27);(1,2)(3,7)(4,5)(6,8)(9,12,13,14)(10,11,15,16)(17,20,21,23)(18,19,22,24)(25,27)(26,28)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fgi,c^2=fgi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4339427);