Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | #15315 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2233 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)(3,6)(4,8)(5,7)(10,14)(11,16)(17,18)(19,22)(20,23)(21,24)(25,26,29,32)(27,30,31,28);(1,3,5,8)(2,6,7,4)(9,10)(11,15)(12,16)(13,14)(17,19)(18,20)(21,23)(22,24)(25,27,29,31)(26,30,32,28);(2,7)(4,6)(10,14)(12,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,30)(26,31,32,27);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,28)(26,27)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,12,13,15)(10,16,14,11)(17,20,21,22)(18,23,24,19)(25,27)(26,28)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4123241);