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 | C2×#20732 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C22×D4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 22 points:
(1,2);(3,4,5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19)(18,21)(20,22);(8,11)(12,14);(16,19)(17,22)(20,21);(7,9,10,13)(8,12,11,14)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#116539, C2×#118020, C2×#6373283, … |
| maximal normal subgroup of | C2×#116539, C2×#118020, C2×#6373283, … |
| maximal quotient of | C2×#358652, #72213483, C2×#494044, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278155);