Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | Q8×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C4⋊C4 |
| prime radical | C23 | C22×Q8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
329 conjugacy classes of subgroups; only the 209 normal subgroups are shown.
This group has 329 conjugacy classes of subgroups — too many to draw. Only its 209 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5);(1,3,4,8)(2,5,6,7);(9,10,13,16,14,17,12,22)(11,18,20,23,21,24,19,15);(9,11,12,19,14,21,13,20)(10,15,16,18,17,23,22,24)70 conjugacy classes — view the full character table.
| Fitting subgroup of | SL(2,𝔽3)×#108 |
| maximal subgroup of | #136349, #136377, #136410, … |
| maximal normal subgroup of | #136349, #136377, #136410, … |
| maximal quotient of | Q8×#334, #226363, Q8×#335, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15720);