Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2332 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,13,12)(11,15,16,14)(19,23)(21,24)(25,26,29,30)(27,28,31,32);(3,4)(6,7)(9,11,13,16)(10,14,12,15)(17,18,20,22)(19,21,23,24)(25,27,29,31)(26,28,30,32);(1,3,5,4)(2,6,8,7)(9,10,13,12)(11,15,16,14)(17,19)(18,21)(20,23)(22,24)(25,28)(26,27)(29,32)(30,31);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,14,16,15)(17,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,31)(30,32);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgi,c^2=fi,d^2=fi,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, 3935993);