Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#3294 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,15)(14,17)(16,18);(3,5)(4,7)(6,10)(8,9);(12,15)(13,18)(16,17);(5,10)(7,9)70 conjugacy classes — view the full character table.
| maximal subgroup of | #220250, #220286, #220362, … |
| maximal normal subgroup of | #220250, #220286, #220362, … |
| maximal quotient of | C2×D4×#102, C2×#22485, C2×#22570, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54100);