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 | #16500 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,13)(11,12,16,15)(17,18,21,19)(20,23,24,22)(25,26,29,32)(27,31,28,30);(2,3)(4,8)(9,11,14,16)(10,15,13,12)(18,19)(20,24)(27,28)(30,31);(4,8)(6,7)(10,13)(11,16)(17,18,21,19)(20,23,24,22)(25,27,29,28)(26,30,32,31);(1,3,5,2)(4,6,8,7)(9,12)(10,11)(13,16)(14,15)(17,19,21,18)(20,22,24,23)(25,28,29,27)(26,31,32,30);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,15,16,12)(17,20,21,24)(18,22,19,23)(25,26)(27,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=fi,d^2=fg,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, 3850473);