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×#3391 |
| 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×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1291 conjugacy classes of subgroups; only the 503 normal subgroups are shown.
This group has 1291 conjugacy classes of subgroups — too many to draw. Only its 503 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);(4,8)(9,10)(11,12)(13,15)(14,17)(16,18);(3,4,7,8)(5,9,6,10);(3,5,7,6)(4,9,8,10)(12,15)(13,18)(16,17);(3,6,7,5)(4,9,8,10)70 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#3391⋊C3 |
| maximal subgroup of | #131062, #131069, #131878, … |
| maximal normal subgroup of | #131062, #131069, #131878, … |
| maximal quotient of | C2×#22540, C2×Q8×#107, C2×#22807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 54375);