Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#3165 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 18 points:
(1,2);(3,4,6,8)(5,9,10,7)(12,15)(13,17)(16,18);(3,5,6,10)(4,7,8,9);(11,12,13,15,14,16,17,18);(12,16)(15,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #112651, #112652, #112658, … |
| maximal normal subgroup of | #112651, #112652, #112658, … |
| maximal quotient of | C2×#20800, C2×#21159, C2×#21322, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53649);