Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#1997 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12,14,17,15,16,18,13);(4,7)(8,9)(11,13,14,16,15,17,18,12);(3,5,6,10)(4,8,7,9)64 conjugacy classes — view the full character table.
| maximal subgroup of | #105768, #105769, #107222, … |
| maximal normal subgroup of | #105768, #105769, #107222, … |
| maximal quotient of | C2×#9878, C2×#9879, #206871, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14323);