Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2027 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
727 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 727 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12)(13,15)(14,17)(16,18);(3,5,6,10)(4,7,8,9)(13,18)(15,16);(12,15)(13,18)(16,17)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137095, #137097, #137123, … |
| maximal normal subgroup of | #137095, #137097, #137123, … |
| maximal quotient of | C2×#10340, C2×#10344, C2×#10343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16447);