Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3283 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(10,13)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26,28,31)(27,30,32,29);(2,5)(3,7)(9,10,12,13)(11,14,15,16)(17,19,20,23)(18,21,22,24)(25,27,28,32)(26,29,31,30);(2,5)(6,8)(10,13)(11,15)(18,22)(21,24)(25,27,28,32)(26,30,31,29);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,29)(28,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,21)(20,23)(22,24)(25,26)(27,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238479, #72263168, #72263224, … |
| maximal normal subgroup of | #72238479, #72263168, #72263224, … |
| maximal quotient of | #72318184 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5156766);