Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C26 |
| direct product | D4×#346 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C8 | C24 |
| commutator | C22 | C22×C4×C8 |
| socle | C23 | C23×C8 |
| prime radical | C22×D4 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4);(2,4);(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(5,6,7,8)(10,14)(15,20)(16,21)(22,24)200 conjugacy classes — view the full character table.
| maximal subgroup of | D8×#346, #72225524, #72225537, … |
| maximal normal subgroup of | D8×#346, #72225524, #72225537, … |
| maximal quotient of | #72225380, D8×#346, #72225524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400407);