Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C43 | C23 |
| socle, prime radical | C23 | C4×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,6)(3,4,7,8)(9,10)(11,13)(12,15)(14,16)(17,18,20,19,21,22,24,23);(1,3,4,6,5,7,8,2)(10,13)(11,16)(14,15);(17,19,20,22,21,23,24,18)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=egh,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,d^b=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4069);