Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#392 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1963 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,21,24,20,22,28,29)(19,25,27,32,26,23,30,31);(1,3)(2,5)(4,8)(6,7)(11,16)(13,15)(18,22)(23,25)(24,29)(31,32);(9,11,12,16)(10,13,14,15)(17,19,20,26)(18,23,22,25)(21,27,28,30)(24,31,29,32);(3,8)(5,7)(10,14)(13,15)(19,26)(23,25)(27,30)(31,32);(2,6)(5,7)(18,22)(23,25)(24,29)(31,32)Polycyclic presentation:
2^9|a^2=h,c^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=dfi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315721);