Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#3184 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
1257 conjugacy classes of subgroups; only the 479 normal subgroups are shown.
This group has 1257 conjugacy classes of subgroups — too many to draw. Only its 479 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 — 5 generators on 18 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(12,15)(13,17)(16,18);(4,7)(8,10)(12,16)(15,18);(11,12,17,18,14,16,13,15);(3,5,6,9)(4,8,7,10)(12,16)(15,18)64 conjugacy classes — view the full character table.
| maximal subgroup of | #111390, #111393, #112637, … |
| maximal normal subgroup of | #111390, #111393, #112637, … |
| maximal quotient of | C2×#20824, C2×#21286, C2×#21410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53679);