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 | |
1974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,8)(3,5,4,7)(9,10,12,14)(11,15,16,13)(18,20)(22,24)(26,29)(27,31);(1,3,6,4)(2,7,8,5)(9,10,12,14)(11,13,16,15)(19,23)(22,24)(25,26,28,29)(27,32,31,30);(9,10)(11,13)(12,14)(15,16)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,29)(31,32);(1,4,6,3)(2,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,24)(21,23)(25,27,28,31)(26,30,29,32);(1,5)(2,4)(3,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,20,21,18)(19,24,23,22)(25,26)(27,32)(28,29)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=gh,d^2=f,e^2=gh,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, 2743730);