Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,8)(9,10,12,13)(11,15,16,14)(17,18,22,24)(19,21,23,20)(25,26,28,30)(27,29,31,32);(1,2,5,6)(3,7,8,4)(10,13)(11,16)(17,19)(18,20)(21,24)(22,23)(27,31)(29,32);(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,23,24,19)(25,26)(27,29)(28,30)(31,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,21,22,20)(18,23,24,19)(25,27,28,31)(26,29,30,32);(1,4,5,7)(2,8,6,3)(9,11,12,16)(10,14,13,15)(17,18)(19,21)(20,23)(22,24)(25,26)(27,29)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=fi,d^2=gh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2952178);