Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #278091 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C22×D4 | C2×D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
556 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)(3,7)(4,6)(5,8)(17,18,20,22)(19,25,26,32)(21,29,27,24)(23,31,30,28);(1,3,5,4)(2,6,8,7)(9,10,11,15,12,14,16,13)(18,22)(19,26)(24,29)(28,31);(1,3,5,4)(2,7,8,6)(10,13)(11,16)(14,15)(17,19,27,31,20,26,21,28)(18,23,29,32,22,30,24,25);(1,4,5,3)(2,7,8,6)(10,14)(13,15)(18,22)(23,30)(24,29)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 232698);