Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 2097152 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26,30,31)(27,32,29,28);(3,7)(6,8)(9,11,12,15)(10,13,14,16)(18,20)(22,23)(25,27)(26,28)(29,30)(31,32);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,20,23)(25,28,30,32)(26,27,31,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,18)(19,22,24,23)(25,29,30,27)(26,28,31,32);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,28)(26,27)(29,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=fg,d^2=f,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135937);