Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393881 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,22,20,24,19,23);(1,2)(3,4)(9,11,12,16)(10,13,14,15)(18,22)(19,21)(23,24);(2,4)(18,22)(19,21)(23,24);(6,8)Polycyclic presentation:
2^9|a^2=ei,b^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72305745, #72305750, #72305791, … |
| maximal normal subgroup of | #72305745, #72305750, #72305791, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303683);