Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#10923 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4090 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,6)(7,8,9,10)(11,12)(13,15)(14,17)(16,18);(12,15,17,16);(3,4,5,6)(8,10)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72209991, C2×#354545, C2×#354635, … |
| maximal normal subgroup of | #72209991, C2×#354545, C2×#354635, … |
| maximal quotient of | C2×#110273, C2×#110284, C2×#110276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261260);