Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,15,16,14)(18,19)(20,24)(26,29)(30,32);(2,3)(4,8)(9,10,13,12)(11,14,16,15)(18,19)(22,23)(27,31)(30,32);(2,3)(6,7)(17,18,21,19)(20,22,24,23)(25,26)(27,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(9,11,13,16)(10,14,12,15)(17,19,21,18)(20,23,24,22)(25,27)(26,30)(28,31)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20,21,24)(18,22,19,23)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=h,d^2=fi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1923985);