Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C210 |
| direct product | C2×#10735 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C27 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C27 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
13194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,15,17,18)(14,16);(4,6)(11,13)(12,15,16,18)(14,17);(8,10)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72218451, #72218744, #72218766, … |
| maximal normal subgroup of | #72218451, #72218744, #72218766, … |
| maximal quotient of | D3×#10735, #72225791, #72231184, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260601);