Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | Q8×#102 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
657 conjugacy classes of subgroups; only the 255 normal subgroups are shown.
This group has 657 conjugacy classes of subgroups — too many to draw. Only its 255 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,10)(7,11,12,9);(5,7,8,12)(6,9,10,11);(1,2,3,4)(13,14)(15,17)(16,19)(18,20);(14,17)(15,20)(18,19)70 conjugacy classes — view the full character table.
| Fitting subgroup of | SL(2,𝔽3)×#102 |
| maximal subgroup of | #220729, #220732, #220738, … |
| maximal normal subgroup of | #220729, #220732, #220738, … |
| maximal quotient of | Q8×D4⋊C8, #224100, #224112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15506);