Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C42 | C24 |
| commutator, socle, prime radical | C23 | C22×C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
680 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,10,14,16,13,15,12,11)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(1,3,6,2)(4,7,5,8)(10,15)(11,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,4,6,5)(2,7,3,8)(9,11,12,15,13,16,14,10)(18,23)(22,28)(24,30)(29,32);(1,5,6,4)(2,7,3,8)(9,12,13,14)(10,11,15,16)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=e,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=deh,c^b=ce152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 91894);