Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
689 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)(3,6)(4,8)(5,7)(9,10,11,13,12,14,15,16)(17,18,22,24,23,19,20,21);(1,3,5,4)(2,6,7,8)(17,19)(18,23)(20,21)(22,24);(1,4,5,3)(2,6,7,8)(17,20,23,22)(18,24,19,21);(17,21,20,19,23,24,22,18)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=f,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,d^c=dh116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 133985);