Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C212 |
| direct product | C2×D4×#95 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
13778 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(16,18);(15,16)(17,18);(8,10)(11,12,14,13);(3,4,5,6)(7,8,9,10)(11,13,14,12)200 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D4×#392, C2×#518019, C2×#4306523, … |
| maximal normal subgroup of | C2×D4×#392, C2×#518019, C2×#4306523, … |
| maximal quotient of | C2×#153827, C2×#154037, #72251086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420666);