Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C8 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C8 | C25 |
| commutator, socle | C22 | C24×C8 |
| prime radical | #537 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1204 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,20,25,21,26,31,39,22,23,32,34,33,35,44,45)(19,27,28,40,29,41,42,48,30,24,36,37,43,38,46,47);(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)(18,23)(24,27)(25,34)(26,35)(37,40)(38,41)(39,45)(47,48);(2,8)(6,9)(7,16)(14,15);(17,19,20,28,21,29,31,42,22,30,32,36,33,43,44,46)(18,24,25,37,26,38,39,47,23,27,34,40,35,41,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,h^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529985);