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×C4 |
| direct product | C22×#1980 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(6,8)(10,13,14,15);(9,10,11,15,12,14,16,13);(5,6,7,8)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#110637, C2×#186250, #72238933, … |
| maximal normal subgroup of | C2×#110637, C2×#186250, #72238933, … |
| maximal quotient of | C2×#375169 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279229);