Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C4×#1887 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4);(6,8)(9,10,14,16,13,11,12,15);(5,6,7,8)(9,11,13,10)(12,15,14,16);(12,14)(15,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72226837, #72226917, #72226934, … |
| maximal normal subgroup of | #72226837, #72226917, #72226934, … |
| maximal quotient of | #72226850, AGL(1,𝔽5)×#1887, #72282907 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261422);