Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×C22⋊C4 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
1794 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)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22);(5,6)(7,8)(9,11)(10,14,16,17)(12,19)(13,18)(15,22,23,24)(20,21);(1,2,3,4)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,g^2=i,b^a=bd,c^a=ce,h^a=hi,c^b=cf,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31210);