Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 524288 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | D4×C4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1537 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,8);(5,6)(7,8);(2,4)(9,10,13,16)(11,15,12,14);(9,11,13,12)(10,14,16,15);(1,2,3,4)(9,12,13,11)(10,15,16,14)70 conjugacy classes — view the full character table.
| maximal subgroup of | #98224, #98252, #130763, … |
| maximal normal subgroup of | #98224, #98252, #130763, … |
| maximal quotient of | D4×#389, D4×#391, C22⋊C4×C4⋊Q8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29108);