Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 1572864 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | Q8×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1133 conjugacy classes of subgroups; only the 635 normal subgroups are shown.
This group has 1133 conjugacy classes of subgroups — too many to draw. Only its 635 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) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(13,14,16,18)(15,20,17,19);(13,15,16,17)(14,19,18,20);(2,4)(5,6,9,12)(7,11,8,10)(13,17,16,15)(14,19,18,20);(5,7,9,8)(6,10,12,11)(13,14,16,18)(15,19,17,20);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,18,16,14)(15,20,17,19)70 conjugacy classes — view the full character table.
| Fitting subgroup of | SL(2,𝔽3)×C4⋊Q8 |
| maximal subgroup of | #98236, #98242, #130937, … |
| maximal normal subgroup of | #98236, #98242, #130937, … |
| maximal quotient of | Q8×#389, Q8×#391, C4⋊C4×C4⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29546);