Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
| direct product | C22×#1586 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C4×D4 |
| prime radical | C23×D8 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2510 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6,8,10)(7,9,12,11)(13,14,16,15,17,18,20,19);(6,9)(7,12)(10,11);(6,10)(9,11)(13,15,16,18,17,19,20,14)152 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#17907, C22×#18001, C22×#18073, … |
| maximal normal subgroup of | C22×#17907, C22×#18001, C22×#18073, … |
| maximal quotient of | C22×#12078, #72286367, C2×#483803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6251737);