Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7654 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,13)(12,15)(14,16)(17,18)(19,23)(20,22)(21,24);(10,13,15,14)(17,19)(18,21,22,23)(20,24);(1,2,3,4)(6,8)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,d^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17559);