Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#388 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
846 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(6,7)(11,17,19,20)(14,22,18,23);(1,2,5,3)(4,6,8,7)(9,10,12,15,13,16,21,24)(11,18,17,23,19,14,20,22);(1,3,5,2)(4,6,8,7)(9,11)(10,14)(12,20)(13,19)(15,23)(16,18)(17,21)(22,24);(1,4,5,8)(2,6,3,7)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=e,f^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,d^c=de,f^c=fi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 287417);