Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,25,20,24,28,32)(19,22,27,29,26,23,30,31);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,20,26)(18,22,24,23)(21,27,28,30)(25,29,32,31);(10,14)(13,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,30)(29,32);(1,3,4,7)(2,6,5,8)(19,26)(22,23)(27,30)(29,31);(18,24)(22,23)(25,32)(29,31)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=di,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278285);