Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | #35051 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C23 ▷ C1 | |
1292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(10,14,15,16)(21,24);(2,5)(3,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24);(9,11,13,12)(10,14,15,16)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=f,c^2=gh,d^2=h,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72267820 |
| maximal normal subgroup of | #72267820 |
| maximal quotient of | #72267814 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4296);