Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| automorphism | ? |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1976 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)(9,10,13,16)(11,15,12,14)(17,18)(19,21)(20,23)(22,24)(25,26)(27,32)(28,30)(29,31);(9,11,13,12)(10,14,16,15)(18,19)(22,23)(26,28)(31,32);(17,18,21,19)(20,22,24,23)(25,27,30,29)(26,31,28,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,15,16,14)(17,19,21,18)(20,23,24,22)(25,28,30,26)(27,32,29,31);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,23)(21,24)(25,29,30,27)(26,32,28,31)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=g,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 797616);