Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7)(3,6,8,5)(9,10)(11,13)(12,14)(15,16)(25,26,28,32)(27,31,29,30);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,18,20,22)(19,23,24,21)(25,27,28,29)(26,30,32,31);(1,3)(2,6)(4,8)(5,7)(10,14)(13,16)(17,19,20,24)(18,21,22,23)(25,27,28,29)(26,31,32,30);(3,8)(5,6)(11,15)(13,16)(26,30)(27,29)(31,32)Polycyclic presentation:
2^9|a^2=e,b^2=gi,c^2=gi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,d^c=de,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386784);