Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #160780 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
1042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(10,14)(11,12)(15,16)(17,18,19,22,20,21,23,24);(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,14,12,16);(18,21)(22,24);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^9|a^2=f,b^2=h,d^2=g,e^2=h,f^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,e^b=eh,f^b=fgh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 67984);