Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | D4×#124 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×D42 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1787 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 16 points:
(6,8);(5,6)(7,8);(2,3)(11,15)(13,16);(1,2,4,3)(9,10,12,14)(11,13,15,16);(1,3,4,2)(9,11)(10,13)(12,15)(14,16)70 conjugacy classes — view the full character table.
| maximal subgroup of | #98258, #130760, #130771, … |
| maximal normal subgroup of | #98258, #130760, #130771, … |
| maximal quotient of | D4×#388, D4×#390, D4×#391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29092);