Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C43 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C24 | #95 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(9,10,13,16)(11,14,15,12)(17,18,20,24,21,23,19,22);(1,2,3,6,4,5,7,8)(9,11,14,10,13,15,12,16)(18,22,23,24);(2,5)(6,8)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=g,e^2=i,f^2=h,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cghi,d^b=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3786);