Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#1642 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#550 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
645 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 143 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) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8)(5,6)(9,10)(11,12,15,18)(13,16,14,17);(3,4)(5,8)(6,10)(7,9)(12,16)(13,14)(17,18);(3,5,7,6)(4,8,9,10)(12,17)(13,14)(16,18)52 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1642, #209705, #209743, … |
| maximal normal subgroup of | C4×#1642, #209705, #209743, … |
| maximal quotient of | C2×#7795, C2×#7774, C2×#7801, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11500);