Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1434 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)(3,7)(4,5)(6,8)(9,10,12,13)(11,15,16,14)(17,18,22,26,21,25,29,32)(19,27,30,24,28,23,20,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(20,30)(23,27)(24,31);(2,5)(6,7)(10,13)(14,15)(17,19)(18,23)(20,29)(21,28)(22,30)(24,32)(25,27)(26,31);(1,3,4,8)(2,6,5,7)(10,13)(14,15)(17,20,21,30)(18,24,25,31)(19,29,28,22)(23,32,27,26);(9,11,12,16)(10,14,13,15)(19,28)(20,30)(23,27)(24,31)Polycyclic presentation:
2^9|a^2=h,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dfi,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316789);