Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
| direct product | C22×#1804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7082 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);(6,8)(9,10,11,12)(13,14)(15,17)(16,18)(19,20);(5,6,7,8)(14,17)(18,19);(14,18)(16,20)(17,19)152 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#24338, C22×#24128, C22×#24196, … |
| maximal normal subgroup of | C22×#24338, C22×#24128, C22×#24196, … |
| maximal quotient of | C2×#392160, C2×#493796, C2×#493794, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277979);