Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| 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 | |
1855 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,16,15)(17,18,22,24)(19,20,23,21)(25,26)(27,30)(28,32)(29,31);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,20,24,21)(25,27,29,32)(26,30,31,28);(1,3,5,8)(2,4,6,7)(9,10,12,14)(11,13,16,15)(17,20,22,21)(18,19,24,23)(25,27)(26,28)(29,32)(30,31);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,20)(21,23)(22,24);(1,4,5,7)(2,3,6,8)(9,11)(10,13)(12,16)(14,15)(17,21,22,20)(18,23,24,19)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1063469);