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×Q8×D8 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1143 conjugacy classes of subgroups; only the 543 normal subgroups are shown.
This group has 1143 conjugacy classes of subgroups — too many to draw. Only its 543 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,5,6)(7,10,9,8);(3,7,5,9)(4,8,6,10);(11,12,13,14,15,16,17,18);(12,18)(13,17)(14,16)70 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×D8×SL(2,𝔽3) |
| maximal subgroup of | #131108, #131117, #131938, … |
| maximal normal subgroup of | #131108, #131117, #131938, … |
| maximal quotient of | C2×Q8×#102, C2×C4⋊C4×D8, C2×Q8×#107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54339);