Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#1300 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | #95 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
393 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 3 generators on 18 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12,15,17,16,13,14,18);(4,7,9,8)(11,13,16,12)(14,17,15,18)76 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#1300, #105422, #105423, … |
| maximal normal subgroup of | C4×#1300, #105422, #105423, … |
| maximal quotient of | C2×#7048, #95347, #95348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1055);