Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#3192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1505 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1505 conjugacy classes of subgroups — too many to draw. Only its 463 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 — 5 generators on 18 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,15)(14,16)(17,18);(3,5)(4,7)(6,10)(8,9)(12,15)(16,17);(5,10)(7,9)(12,16)(14,18)(15,17);(4,8)(7,9)(12,15)(16,17)52 conjugacy classes — view the full character table.
| maximal subgroup of | #111406, #111452, #111479, … |
| maximal normal subgroup of | #111406, #111452, #111479, … |
| maximal quotient of | C2×#21764, C2×#21942, C2×#22046, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53723);