Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
| direct product | C22×#1792 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
19058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(1,2);(3,4);(5,6,7,8)(10,13)(11,15)(14,16);(9,10)(11,16)(12,14)(13,15);(11,15)(13,16)128 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#13343, #72233101, C2×#500009, … |
| maximal normal subgroup of | C22×#13343, #72233101, C2×#500009, … |
| maximal quotient of | #72232846, C2×#186594, C2×#186250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277936);