Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×#1294 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
727 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2);(3,4,5,6)(7,8)(9,11)(10,13)(12,14)(15,16)(17,21)(18,20)(19,22);(8,11,13,12)(15,17)(16,19,20,21)(18,22)76 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1294, #105409, #105410, … |
| maximal normal subgroup of | C4×#1294, #105409, #105410, … |
| maximal quotient of | C2×#7035, #95189, #95206, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 999);