Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162118 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2856 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1034 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,6,3,8)(9,10,13,15,14,11,12,16)(17,18,21,24,22,19,20,23);(2,6)(3,4)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20,22,21)(18,23,19,24);(2,6)(7,8)(17,20,22,21)(18,24,19,23)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cg,d^a=dgh,e^a=eg,c^b=ci,e^b=eh,f^b=fgh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 69513);