Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
| direct product | Q8×SD32 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
247 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 247 conjugacy classes of subgroups — too many to draw. Only its 117 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:
(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24);(10,16)(11,23)(12,14)(13,21)(15,19)(18,24)(20,22)55 conjugacy classes — view the full character table.
| Fitting subgroup of | SL(2,𝔽3)×SD32 |
| maximal subgroup of | #491841, #491842, #491848, … |
| maximal normal subgroup of | #491841, #491842, #491848, … |
| maximal quotient of | Q8×#357, Q8×#358, C4⋊C4×SD32, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26934);