Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#2050 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #124 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | #394 |
| prime radical | C2×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
691 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 691 conjugacy classes of subgroups — too many to draw. Only its 167 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) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,12)(6,14)(7,15)(8,11)(9,13)(10,17)(16,18);(3,5)(4,9)(6,10)(7,16)(8,14)(11,18)(12,15)(13,17);(3,6,7,9,8,12,17,18)(4,5,10,13,11,14,15,16)52 conjugacy classes — view the full character table.
| maximal subgroup of | #137199, #137200, #137201, … |
| maximal normal subgroup of | #137199, #137200, #137201, … |
| maximal quotient of | C2×#10518, C2×#10520, C2×#10522, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16514);