Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,15)(12,16)(13,14)(18,19)(22,23)(25,26,28,30)(27,31,32,29);(1,2,4,5)(3,8,7,6)(9,10,13,14)(11,15,16,12)(17,18,21,19)(20,23,24,22)(27,32)(29,31);(3,7)(6,8)(10,14)(12,15)(17,18,21,19)(20,23,24,22)(25,26)(27,29)(28,30)(31,32);(1,2)(3,8)(4,5)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,18)(20,22,24,23)(25,27)(26,29)(28,32)(30,31);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,20,21,24)(18,22,19,23)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fgi,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3919660);