Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
| direct product | C22⋊C4×M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator | C22 | C22×C42 |
| socle | C23 | C23×C4 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6,7,8);(2,4);(9,10,11,14,12,13,15,16);(10,13)(14,16)100 conjugacy classes — view the full character table.
| maximal subgroup of | #109633, #109654, #110090, … |
| maximal normal subgroup of | #109633, #109654, #110090, … |
| maximal quotient of | #181932, M16×C22⋊C8, #183039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13376);