Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C16 | C24 |
| commutator | C2 | C23×C32 |
| socle | C22 | C23×C16 |
| prime radical | C23 | C22×C16 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,d^a=de,c^b=ce272 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420002);