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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,12,13)(11,15,16,14)(17,18)(19,21)(20,23)(22,24)(25,26,29,30)(27,28,32,31);(4,8)(6,7)(10,13)(14,15)(17,18,21,19)(20,23,24,22)(25,27)(26,28)(29,32)(30,31);(2,3)(6,7)(10,13)(11,16)(18,19)(20,24)(26,30)(27,32);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,19,21,18)(20,22,24,23)(25,27)(26,31)(28,30)(29,32);(1,4,5,8)(2,6,3,7)(9,10)(11,15)(12,13)(14,16)(17,20,21,24)(18,22,19,23)(25,28,29,31)(26,27,30,32)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,d^2=fgh,e^2=ghi,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, 4206588);